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+{"text":"\\section{Introduction}\n\nA {\\em Margulis spacetime\\\/} is a geodesically complete flat \nLorentzian $3$-manifold $M^3$ with free fundamental group.\nSuch manifolds are quotients $\\Eto\/\\Gamma$ of \n$3$-dimensional {\\em Minkowski space\\\/} $\\Eto$ by a discrete group $\\Gamma$\nof isometries acting properly on $\\Eto$. \nBy\n\\cite{FriedGoldman,Mess} every complete affinely \nflat $3$-manifold\nhas solvable fundamental group\nor is a  Margulis spacetime.\nIn the latter case the linear holonomy \n$$\n\\pi_1(M) \\cong \\Gamma\\xrightarrow{\\LL} {\\o{SO}(2,1)}\n$$ \nis an embedding of $\\Gamma$ onto a discrete subgroup\n$\\Gamma_0  = \\LL(\\Gamma)$.\nThus, associated to every Margulis spacetime is a complete hyperbolic\nsurface $\\surf = \\hp\/\\Gamma $. \nThis hyperbolic surface has an intrinisic description:\nit consists of parallelism classes of timelike lines \n(particles) in $M$.  \nIn particular $\\Gamma$ is an {\\em affine deformation\\\/} of the\nFuchsian group $\\Gamma_0$, and we say that $M$ is an \n{\\em affine deformation\\\/}\nof the hyperbolic surface $\\surf$. \n\nWe conjecture that every Margulis spacetime $M$ is {\\em tame},\nthat is, admits a polyhedral decomposition by crooked planes.\nA consequence is that $M$ is {\\em topologically tame,\\\/} that is,\nhomeomorphic to an open solid handlebody.\nWe start with the simplest groups of interest,\nthose whose holonomy is a rank two group.  (The cyclic case is\ntrivial.)  We previously established the conjecture when $\\surf$ is a\nthree-holed sphere~\\cite{CDG}.  In this paper, we establish the\nconjecture for its nonorientable counterpart, when $\\surf$ is\nhomeomorphic to a two-holed projective plane.  Every nonorientable\nsurface of negative Euler characteristic contains an embedded\ntwo-holed cross-surface.  For this reason, we expect the two-holed\ncross-surface, like the three-holed sphere, to act as a building block\nfor proving the general conjecture.\n\nJ.\\ H.\\ Conway has introduced the term {\\em cross-surface\\\/} for the\ntopological space underlying the real projective plane $\\R\\mathbf{P}^2$.  Let\n$\\topsurf$ be a {\\em $2$-holed cross-surface,\\\/} that is, the topological\nsurface underlying the complement of two disjoint discs in $\\R\\mathbf{P}^2$.\nAs is common in deformation theory, we fix the topology\nof $\\topsurf$ and consider {\\em marked geometric structures,\\\/}\nthat is, homotopy classes of homeomorphisms $\\topsurf\\rightarrow\\surf$,\nwhere $\\surf$ is a surface with a hyperbolic structure.\n\nThe first part of the paper focuses on the hyperbolic structure.  In\nparticular we describe the space of marked complete hyperbolic\nstructures on $\\topsurf$.  We call this space the {\\em Fricke space}\nof $\\topsurf$ and denote it by $\\mathfrak{F}(\\topsurf)$.  In the second part of the\npaper, we explicitly describe the space of all Margulis spacetimes\narising from $\\topsurf$ as a bundle of convex $4$-sided cones over\n$\\mathfrak{F}(\\topsurf)$.\n\nLike~\\cite{CDG}, the classification involves both the topology of $M$ and the\ngeometry of the deformation space.\nWe show that every affine deformation of $\\surf$ is homeomorphic to a solid\nhandlebody of genus two by constructing an explicit fundamental polyhedron.  \nAlthough the two-holed projective plane is a nonorientable\nsurface, its affine deformations are orientable  $3$-manifolds.\nIn fact, every Margulis spacetime is orientable (Lemma~\\ref{lem:orientable}).\n\nThere are four isotopy classes of \nessential primitive simple closed curves on $\\topsurf$.\nTwo of these curves, denoted $A$ and $B$, \ncorrespond to components of $\\partial\\topsurf$. \nThe other two, denoted $X$ and $Y$, \nreverse orientation and intersect transversely in one point.\n\nIf we fix a hyperbolic structure $\\surf$, \nthe boundary curves $A$ and $B$ correspond either to\nclosed geodesics bounding {\\em funnels\\\/} (complete ends\nof infinite area with cyclic fundamental group) or {\\em cusps\\\/} \n(ends of finite area).\nWhen a curve $K$ bounds a funnel, \n$\\ell_K$ is the (positive) length of the closed geodesic and \nwhen the boundary curve cuts off a cusp $\\ell_K$ is set equal to $0$. \nAn orientation-reversing curve  $K$ always corresponds to a closed geodesic, \nso that $\\ell_K$ represents this length and is always positive.\nFor\n$$\n\\big( \\ell_A, \\ell_B, (\\ell_X,\\ell_Y)\\big) \n\\;\\in\\;\n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+^2\n$$\nthe lengths satisfy the identity\n\\eqref{eq:LengthIdentity}, and the Fricke space $\\mathfrak{F}(\\topsurf)$\nidentifies with \n$$\n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+ .\n$$\nThe boundary $\\partial\\mathfrak{F}(\\topsurf)$ has two faces sharing a common edge.\nOne face is defined by $\\ell_A = 0$ and the other by $\\ell_B = 0$.\nThese faces intersect in the line $\\ell_A = \\ell_B = 0$, which\ncorresponds to complete hyperbolic structures with finite area.\n\n\\medskip\n\nFix a linear representation $\\rho_0$\nof $\\pi_1(S)$ in the special orthogonal group ${\\o{SO}(2,1)}$.\nAn {\\em affine deformation\\\/} of $\\rho_0$  \nis a representation $\\rho$ in the group of Lorentzian isometries\nwhose linear part is $\\rho_0$, that is, \n\\begin{equation*}\n\\LL\\circ\\rho = \\rho_0.\n\\end{equation*}\nAn affine deformation $\\rho$ of $\\rho_0$ \nis {\\em proper\\\/} if it defines a proper affine action\nof $\\pi_1(S)$ on $\\Eto$.\nEquivalently $\\rho$ is the holonomy representation of a complete\naffine structure on a $3$-manifold $M^3$ and the isomorphism\n$$\n\\pi_1(S) \\longrightarrow \\pi_1(M^3)\n$$\ninduced by $\\rho_0$ and $\\rho$\narises from a homotopy-equivalence \n$ S \\longrightarrow M^3$.\n\nAffine deformations of the fixed linear representation $\\rho_0$\ncorrespond to {\\em cocycles\\\/} and form a vector space $Z^1(\\Gamma_0,\\rto)$.\nSlightly abusing notation, denote also by $\\mathbb{V}$ the $\\Gamma_0$-module\ngiven by the linear holonomy representation $\\rho_0$ of $\\Gamma_0$ on\nthe vector space $\\mathbb{V}$.  Translational conjugacy classes of affine\ndeformations form the {\\em cohomology group\\\/}\n$H^1\\big(\\Gamma_0,\\mathbb{V}\\big)$.\n\nWhen $\\surf$ is a two-holed cross-surface,\nor, more generally, any surface with $\\pi_1(\\Sigma)$ free\nof rank two,  $H^1(\\Gamma_0,\\rto) \\cong \\mathbb R^3$.\nIn his original work~\\cite{Margulis1,Margulis2}, \nMargulis introduced a \n{\\em marked signed Lorentzian length spectrum invariant,\\\/}\nwhich now bears his name. This $\\mathbb R$-valued class function on $\\pi_1(\\surf)$\ndetects the properness of an affine deformation, and determines an affine\ndeformation up to conjugacy~\\cite{CharetteDrumm_isospectrality,DrummGoldman_isospectrality}.\n\nUnder the correspondence between\naffine deformations and infinitesimal deformations of the hyperbolic structure on\n$\\hp$, Margulis's invariant identifies with the first derivative of the \ngeodesic length function on Fricke space \n\\cite{GoldmanMargulis,Goldman_MargulisInvariant}.\nOriginally defined for hyperbolic\nisometries of $\\Eto$, it has been extended to parabolic isometries \nin \n\\cite{CharetteDrumm}, and to a function on geodesic currents \nin\n\\cite{GLM}. \n\nGiven a curve $K\\subset\\surf$, the Margulis invariant $\\mu_K$ is a linear\nfunctional on $H^1(\\Gamma_0,\\rto)$.  The four functionals associated to the curves\n$A,B,X,Y$ satisfy the linear dependence \\eqref{eq:alphaCDMrelations}.\nLet $\\Dd$ denote the open cone in $H^1(\\Gamma_0,\\rto)$ such that all the quantities $\n\\tilde{\\alpha}_A, \\;\\tilde{\\alpha}_B , \\;\\tilde{\\alpha}_X , \\;\\tilde{\\alpha}_Y.  $ are either all positive or all\nnegative.  Figure~\\ref{fig:DeformationSpace} offers a projective view\nof $\\Dd$; each line corresponds to a zero-set of a $\\mu_K$ for some\ncurve $K$. \n\nProper affine deformations will be investigated via \n{\\em crooked  planes\\\/}, polyhedral surfaces introduced \nin\n\\cite{Drumm_thesis,Drumm_jdg,DrummGoldman_crooked}.  \nA proper affine deformation is tame if it admits a fundamental\npolyhedron bounded by crooked planes.  \nThe {\\em Crooked Plane Conjecture\\\/} \n\\cite{DrummGoldman_crooked,CDG}, which we prove for affine\ndeformations of the two-holed cross-surface, asserts that every\nproper affine deformation is tame.\n\nDenote the subspace of $H^1(\\Gamma_0,\\rto)$ consisting of proper affine\ndeformations by $\\Pr$ and the subset of $\\Pr$ consisting \nof tame affine deformations by $\\Ta$. \nOur main result is that $$\\Ta \\;=\\; \\Pr\\;=\\Dd.$$\n\n\n\\begin{thmmain}\nLet $\\surf$ be a two-holed cross-surface.  Then every proper affine deformation of $\\surf$ is tame and the affine $3$-manifold is homeomorphic to a handlebody\nof genus two.\n\\end{thmmain}\n\nLet us underscore the fact that $\\Pr$ is finite-sided, just like in the\ncase of the three-holed sphere, where proper deformations make up\na three-sided polyhedral cone~\\cite{CDG}. In all other cases, the space of\nproper affine deformations has infinitely many faces. \nIn particular, for the other two surfaces of Euler characteristic\n$ -1$ (or equivalently, surfaces with fundamental group of rank two), \nnamely the one-holed torus and one-holed Klein bottle,\nthe space of proper affine deformations is defined by (necessarily)\ninfinitely many linear conditions~\\cite{Charette_nonproper,CDGp}.\n\n\\subsection*{Acknowledgments}\nWe are grateful to Francis Bonahon, Suhyoung Choi, David Gabai, \nFran\\c cois Labourie, Grisha Margulis and  Yair Minsky, \nSer-Peow Tan for helpful conversations. \nWe also thank the referee for many useful suggestions.\n\n\\section{The two-holed cross-surface}\\label{sec:thx}\n\nIn this first section, we investigate the topology of a two-holed cross-surface.  Then we endow it with a hyperbolic structure.\n\n\\subsection{Curves and the fundamental group of $\\topsurf$}\n\nWe begin by reviewing the topology of the two-holed cross-surface\n$\\topsurf$.  Recall that $X$ and $Y$ are orientation-reversing curves and that $A$ and $B$ bound $\\topsurf$. \n\nRepresent $\\topsurf$ as the nonconvex component\nof the complement of two disjoint discs in $\\R\\mathbf{P}^2$ as follows. \nThe coordinate axes in $\\mathbb R^2$ extend to projective lines in $\\R\\mathbf{P}^2$\nwhich intersect exactly once (at the origin) in $\\mathbb R^2$. Choose two disjoint\nhyperbolas in $\\mathbb R^2$ which intersect the ideal line in pairs of points\nwhich do not separate each other. In $\\R\\mathbf{P}^2$ these hyperbolas extend to\nconics which bound two disjoint convex regions. The complement of these\nconvex regions is a model for $\\surf$.\nIn  Figure~\\ref{fig:hyperbolas}, the complement of $X$ and $Y$  in $\\surf$ is \nfoliated by curves homotopic to the boundary curves $A$ and $B$.\n\nChoosing orientations and arcs from a basepoint  \n$\\basepoint$ to each of these four curves,\ndefine elements of $\\fg$ corresponding to $A, B, X, Y$. \nDenoting these elements also by $A,B,X,Y$, respectively, \nyields the redundant presentation\n\\begin{equation}\\label{eq:redundant}\n\\fg \\;=\\; \\langle A, B, X, Y \\mid X Y = A, \\, \\bar{Y} X = B \\rangle\n\\end{equation}\nNote that any two-element subset of $\\{A,B,X,Y\\}$ \n{\\em except\\\/} $\\{A,B\\}$ freely generates $\\fg$\nand \n\\begin{equation}\nA B = X^2.\n\\end{equation}\nSimple closed curves on $\\surf$ are easily classified. \n(See for example\n\\cite{Bonahon}.)\nIf $\\gamma\\subset S$ is a simple closed curve, then exactly one of the following holds:\n\\begin{itemize}\n\\item $\\gamma$ is contractible and  bounds a disc;\n\\item $\\gamma$ is peripheral and is isotopic to either $A$ or $B$;\n\\item $\\gamma$ is orientation-reversing and is isotopic to either $X$ or $Y$;\n\\item $\\gamma$ is essential, nonperipheral, orientation-preserving and is isotopic to either $X^2$ or $Y^2$.\n\\end{itemize}\n\n\\setcounter{figure}{0}\n\\begin{figure}[b]\n\\includegraphics[scale=1.0\n{hyperbolas4.eps}\n\\caption{The complement of two disjoint discs in \n$\\R\\mathbf{P}^2$ is homeomorphic to a $2$-holed cross-surface.\nThe coordinate axes correspond to the two orientation-reversing \nsimple loops $X$ and $Y$.\nThe two hyperbolas $A$ and $B$ define components of the boundary.}\n\\label{fig:hyperbolas}\n\\end{figure}\n\n\\subsection{Hyperbolic structures on $\\topsurf$}\nConsider a complete hyperbolic surface $S$ and a diffeomorphism (the {\\em  marking\\\/})\n$\\Sigma\\longrightarrow S$. \nChoose a universal covering $\\tS\\to S$, \nand a lift $\\tbasepoint\\in\\tS$ of the basepoint $\\basepoint\\in\\surf$.\nChoose a developing map\n$\\tS\\xrightarrow{\\mathsf{dev}} \\hp$ and holonomy representation \n$$\n\\fg \\xrightarrow{\\rho_0} \\Gamma \\subset \\mathsf{Isom}(\\hp)\\cong\\mathsf{PGL}(2,\\R).\n$$\nThe holonomy representation embeds $\\fg$ as a discrete subgroup\n$\\Gamma_0 = \\rho_0(\\fg)$.\nThe hyperbolic surface $S$ may have finite or infinite area. \n\nWhen $\\surf$ is nonorientable,\nsome elements of $\\Gamma$  reverse orientation.\nWhen $S$ is a two-holed cross-surface as above,\n$\\rho_0(X)$ and $\\rho_0(Y)$ are glide-reflections, \nwhile $\\rho_0(A)$ and $\\rho_0(B)$ are either transvections\nor parabolic isometries of $\\mathsf{H}^2$.\n(Compare Figure~\\ref{fig:FundamentalDomain}.)\n\nHenceforth, when the context is unambiguous,\nwe suppress $\\rho_0$, for example, simply writing $A$ for $\\rho_0(A)$.\n\n\\subsection{Fundamental domains for the two-holed cross-surface}\n\\label{sec:funddomainsHypSurf}\n\nWe now\nconstruct a fundamental domain for the action of the associated group\n$\\Gamma_0$ on the {\\em Nielsen convex region,\\\/} a convex\n$\\Gamma_0$-invariant open subset $\\Omega \\subset\\hp$.  Fundamental\ndomains for incomplete surfaces give rise to fundamental domains\nin $\\Eto$ for complete Margulis space-times.\n\n\nDenote the attracting and repelling fixed points of a hyperbolic\nisometry $W$ of $\\hp$ by $\\xp{W}$ and $\\xm{W}$ respectively.  \nIf $W$ is parabolic \nthen either annotation represents the fixed point on $\\partial\\hp$.\nAssume the fixed points \n\\begin{equation*}\n\\xp{X},\\xp{Y},\\xm{X},\\xm{Y}\n\\end{equation*}\nare in counter-clockwise order. \n(Compare Fig.\\ref{fig:FundamentalDomain}.)\nIn $\\hp$, let $\\mathcal Q$ be the ideal quadrilateral \nwith vertices:\n\\begin{equation*}\n\\xp{B},\\xm{A}, A(\\xp{B}), X(\\xp{B})\n\\end{equation*}\n(Compare Figure~\\ref{fig:FundDomainIdentifiedN1}.)\nThe transvection $A$ and the glide-reflection $X$ identify the four sides\nof $\\mathcal Q$ as follows:\n\\begin{align*}\n\\Big(\\xm{A},\\,\\xp{B}\\Big)     &\\;\\stackrel{A}\\longmapsto\\; \\Big(\\xm{A},\\, A(\\xp{B})\\Big) \\\\\n\\Big(\\xp{B},\\, X(\\xp{B})\\Big) &\\;\\stackrel{X}\\longmapsto\\; \\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)\n\\end{align*}\nsince $A(\\xp{B}) = X^2(\\xp{B})$.\n(Compare Figure~\\ref{fig:FundDomainIdentifiedN1}.)\nThe diagonal\n$\\Big(\\xm{A},\\, X(\\xp{B})\\Big)$ divides $\\mathcal{Q}$\ninto two ideal triangles. \n(Compare Figure~\\ref{fig:IdealQuadrilateral}.)\nDenote the interior of the complement of a halfplane\n$\\mathfrak{H}$ by $\\mathfrak{H}^c$.\nThe sides of $\\mathcal Q$ bound four disjoint halfplanes which are pairwise\nidentified by $A$ and $X$ to define a Schottky-like system:\n\\begin{itemize}\n\\item\nThe side $\\Big(\\xm{A},\\, A(\\xp{B})\\Big)$ bounds a halfplane $\\mathfrak{H}_A$; \n\\item\nThe transvection $\\bar{A}$ maps \n$\\Big(\\xm{A},\\, A(\\xp{B})\\Big)$ \nto the side $\\Big(\\xm{A},\\,\\xp{B}\\Big)$\nbounding the halfplane $\\bar{A}(\\mathfrak{H}_A^c)$.\n\\item The side \n$\\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)$ bounds a halfplane $\\mathfrak{H}_X$;\n\\item\nThe glide-reflection $\\bar{X}$ maps \n$\\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)$ to the side\n$\\Big(\\xp{B},\\, X(\\xp{B})\\Big)$ bounding the halfplane $\\bar{X}(\\mathfrak{H}_A^c)$.\n\\item\nThe four halfplanes \n$$\n\\mathfrak{H}_A, \\bar{A}(\\mathfrak{H}_A^c), \\mathfrak{H}_X, \\bar{X}(\\mathfrak{H}_A^c)\n$$ \nare pairwise disjoint.\n\\end{itemize}\nThe fundamental domain and its corresponding set of identifications for the\nhyperbolic surface $S = \\Omega\/\\Gamma_0$ \nserve as a template for the crooked fundamental domains constructed in\n\\S\\ref{sec:CrookedFD}. The positions of the halfplanes and their bounding\ngeodesics  play a crucial role in ensuring disjointness of the crooked \nplanes.\n\\newpage\n\n\\begin{figure}[t]\n\\psfrag{m}{$X^-$}\n\\psfrag{V}{$X^+$}\n\\psfrag{Z}{$Y^-$}\n\\psfrag{W}{$Y^+$}\n\\psfrag{X}{$X$}\n\\psfrag{Y}{$Y$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$B$}\n\\psfrag{P}{$A^Y\\,=\\,A^{\\bar{X}}$\n\\psfrag{Q}{\\qquad\\qquad$B^Y\\,=\\,B^X$\n\\psfrag{o}{$\\mathsf{dev}(\\tbasepoint)$}\n\\includegraphics[scale=1.1]\n{c02b1.eps}\n\\caption{\nA fundamental domain for the two-holed cross-surface $\\surf$.\n}\n\\label{fig:FundamentalDomain}\n\\end{figure}\n\n\\begin{figure}[h]\n\\psfrag{X}{$X$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$\\xp{B}$}\n\\psfrag{C}{$\\xm{A}$}\n\\psfrag{D}{$A(\\xp{B})$}\n\\psfrag{E}{$X(\\xp{B})$}\n\\includegraphics[scale=1.]\n{FundDomainIdentifiedN1.eps}\n\\caption{Identifications of the ideal quadrilateral.\nThe transvection $A$ identifies the bottom two sides,\nwhich share the ideal vertex $\\xm{A}$.\nThe glide-reflection $X$ identifies the top two\nsides, which share the ideal vertex $X(\\xp{B})$.\nThe arrows on the sides indicate how to identify the\nsides.\n}\n\\label{fig:FundDomainIdentifiedN1}\n\\end{figure}\n\n\\begin{figure}[h]\n\\psfrag{A}{$\\bar{X}(\\HX^c)$}\n\\psfrag{B}{$\\xp{B}$}\n\\psfrag{C}{$\\bar{A}(\\HA^c)$}\n\\psfrag{E}{$\\xm{A}$}\n\\psfrag{F}{$\\mathfrak{H}_A$}\n\\psfrag{G}{$A(\\xp{B})$}\n\\psfrag{H}{$\\mathfrak{H}_X$}\n\\psfrag{I}{$X(\\xp{B})$}\n\\includegraphics[scale=1.11]\n{IdealTriShade.eps}\n\\caption{The ideal quadrilateral $\\mathcal Q$ with diagonal,\nand the four halfplanes $\\mathcal Q$ bounds. \n}\n\\label{fig:IdealQuadrilateral}\n\\end{figure}\n\n\n\n\n\n\n\\section{The Fricke space of $\\topsurf$}\nThis section describes the deformation space of\nmarked hyperbolic structures on $\\topsurf$ in\nterms of representations and their characters in $\\SL(2,\\C)$.\nThen we find a simple set of coordinates for $\\mathfrak{F}(\\topsurf)$\nin terms of traces of $2\\times 2$ real matrices.\n\n\n\\subsection{Representing orientation-reversing isometries by matrices}\nFor the following discussion of the Fricke space of the two-holed\ncross-surface, represent $\\hp$ as a totally geodesic hypersurface in\nhyperbolic $3$-space $\\hthree$.\n{\\em Every\\\/} isometry of $\\hp$ extends uniquely to an \n{\\em orientation-preserving\\\/} isometry of $\\hthree$ \npreserving $\\hp\\subset\\hthree.$\n(Compare \\cite{Goldman,Marden}.\nTo use the algebraic machinery of traces in $\\SL(2,\\C)$, \nwe must also lift $\\rho_0$ to a representation $\\trho_0$ in the double\ncovering $\\SL(2,\\C)$ of the identity component \n$$\n\\mathsf{Isom}^+(\\hthree)\\cong\\mathsf{PSL}(2,\\C).\n$$\nSuch a lift is always possible since $\\fg$ is a free group.\nDenote the lifted elements of $\\SL(2,\\C)$ by $A, B, X, Y$ respectively, \nas well.\n\nFollowing  \\cite{Goldman}, represent a lift of an isometry of $\\hp$ as an element\nof $\\SL(2,\\C)$ which is real if the isometry preserves orientation on $\\hp$, \nand purely imaginary it the isometry reverses orientation $\\hp$. \nIn the real case, the matrix lies in $\\SL(2,\\R)$, and is defined up to $\\pm 1$.\nIt may be elliptic, parabolic or hyperbolic. If it is parabolic, then it is conjugate\nto:\n\\begin{equation}\\label{eq:Unipotent}\n\\pm \\bmatrix 1 & 1 \\\\ 0 & 1 \\endbmatrix\n\\end{equation}\nIf it is hyperbolic then it is conjugate to the diagonal matrix:\n\\begin{equation}\\label{eq:Transvection}\n\\pm \\bmatrix e^{\\ell\/2}  & 0 \\\\ 0 & e^{-\\ell\/2} \\endbmatrix\n\\end{equation}\nwith trace $\\pm 2 \\cosh(\\ell\/2)$. \nIt corresponds to a {\\em transvection\\\/} of $\\mathsf{H}^2$.\nIt leaves invariant a geodesic (its {\\em invariant axis),\\\/}\nwhich for the above example corresponds to the imaginary axis in the upper halfplane model, and it displaces points on its axis by length $\\ell$.\n\nIn the purely imaginary case, \nthe corresponding purely imaginary element of $\\SL(2,\\C)$ is $i P$, \nwhere $P\\in\\mathsf{GL}(2,\\R)$ and $\\det(P)\\,= \\, -1$.\nFor example the diagonal matrix\n\\begin{equation}\\label{eq:GlideReflection}\niP = i \\bmatrix e^{\\ell\/2 } & 0 \\\\ 0 & -e^{-\\ell\/2} \\endbmatrix \n\\;\\in\\; \\SL(2,\\C)\n\\end{equation}\nrepresents a glide-reflection of displacement length $\\ell$ along the \ngeodesic in the\nupper halfplane represented by the imaginary axis. \nIts trace equals $2i\\sinh(\\ell\/2)$.\nSince a matrix in $\\SL(2,\\C)$ representing an isometry of hyperbolic space \nis only determined up to multiplication by $\\pm 1$, \nthe matrix in $\\SL(2,\\C)$ representing a glide-reflection of\ndisplacement length $\\ell$ has trace $\\pm 2i \\sinh(\\ell\/2)$.\n\n\\subsection{Trace coordinates}\nSuppose that $\\trho_0$ is a representation in $\\SL(2,\\C)$ \n(preserving ${\\mathsf H}^2\\subset\\hthree$) which covers\na holonomy representation of a marked hyperbolic structure\non $\\Sigma$. \n\nThe character of $\\trho_0$ corresponds to a quadruple\n$(a,b,x,y)\\in\\mathbb R^4$ defined by:\n\\begin{align*}\na &:= \\mathsf{tr}(A) \\\\\nb &:= \\mathsf{tr}(B) \\\\\nx &:= -i\\ \\mathsf{tr}(X)  \\\\\ny &:= -i\\  \\mathsf{tr}(Y) \n\\end{align*}\nsubject to the trace identity\n\\begin{equation}\\label{eq:trace_identity}\na + b + x y \\;=\\; 0,\n\\end{equation}\nwhich arises directly from the ``Basic Trace Identity'' in \\cite{Goldman}:\n\\begin{equation*}\n \\mathsf{tr} (XY) + \\mathsf{tr} (X\\bar{Y}) = \\mathsf{tr} (X) \\mathsf{tr} (Y)\n\\end{equation*}\n\\big(which in turn is just the Cayley-Hamilton theorem for $\\SL(2,\\C)$\\big).\n\nThe {\\em character set\\\/}\n\\begin{equation}\\label{eq:CharacterSet}\n\\mathscr{C} :=\n\\{ (a,b,x,y)\\in\\mathbb R^4 \\mid   a + b = - x y,\\ \\vert a\\vert \\ge 2, \\vert b\\vert \\ge 2 \\}\n\\end{equation}\nidentifies with conjugacy classes of lifts to $\\SL(2,\\C)$ of holonomy representations\nof marked complete hyperbolic structures on $\\Sigma$.\n\n\n\nSince $\\{X,Y\\}$ freely generates $\\fg$, different lifts $\\trho_0$ of $\\rho_0$\ndiffer by {\\em multiplication by a character\\\/}\n$$\n\\fg\\xrightarrow{\\chi}\\{\\pm 1\\}\n$$\nas follows. \nThe group \n$$\n\\mathscr{G}\\ :=\\  \\mathsf{Hom}\\big(\\fg,\\{\\pm 1\\}\\big)\\ \\cong\\  \\mathbb Z\/2 \\oplus \\mathbb Z\/2\n$$ \nacts on $\\mathsf{Hom}(\\fg,\\SL(2,\\C))$ by pointwise multiplication:\n$$\n\\gamma \\stackrel{\\chi\\cdot\\rho}\\longmapsto \\chi(\\gamma)\\rho(\\gamma)\n$$\nwhich is a representation since $\\{\\pm 1\\} \\subset\\SL(2,\\C)$ is central.\n\nOn the quotient of $\\mathsf{Hom}\\big(\\fg,\\SL(2,\\C)\\big)$ \nby $\\mathsf{Inn}\\big(\\SL(2,\\C)\\big)$, \nthe induced $\\mathscr{G}$-action\nis described by the action on traces: \n\\begin{align*}\n(a,b,x,y) &\\mapsto (a,b,-x,-y); \\\\\n(a,b,x,y) &\\mapsto (-a,-b,-x,y); \\\\\n(a,b,x,y) &\\mapsto (-a,-b,x,-y).\n\\end{align*}\nChanging the sign of either $\\mathsf{tr}(X)$ or $\\mathsf{tr}(Y)$ results in changing the sign of $\\mathsf{tr}(A)$ and $\\mathsf{tr}(B)$, \nwhile changing the signs of both $\\mathsf{tr}(X)$ and $\\mathsf{tr}(Y)$ \nleaves the signs of $\\mathsf{tr}(A)$ and $\\mathsf{tr}(B)$ unchanged. \n\nThe Fricke space $\\mathfrak{F}(\\topsurf)$ then identifies with the quotient \n$\\mathscr{C}\/\\mathscr{G}$.\nSince none of  $a,b,x,y$ vanish, the action of $\\mathscr{G}$ is free and $\\mathfrak{F}(\\topsurf)$ also \nidentifies with one of the four connected components of the character set.\n\n\\subsection{Relating traces to lengths}\nThe nonperipheral essential orien\\-ta\\-tion-reversing simple curves $X$ and $Y$ are uniquely\nrepresented by closed geodesics. Similarly the boundary curves $A$ and $B$ are closed\ngeodesics or cusps. Denote the lengths of these geodesics by $\\lX, \\lY,\\lA, \\lB$\nrespectively. Denote the angle of intersection of the geodesic representatives $X$ and $Y$\nby $\\theta$. Explicit matrix representatives are given below:\n\n\\begin{align*}\nX& \\longleftrightarrow \ni\\bmatrix \n\\sinh\\frac{\\lX}{2} + \\cos\\theta \\cosh\\frac{\\lX }{2}  &\n\\sin \\theta  \\cosh\\frac{\\lX}{2}  \\\\\n\\sin\\theta \\cosh\\frac{\\lX}{2} & \n\\sinh\\frac{\\lX}{2} - \\cos\\theta \\cosh\\frac{\\lX}{2} \n\\endbmatrix \\\\\nY & \\longleftrightarrow i\\bmatrix  e^{\\lY\/2 } & 0 \\\\ 0 & -e^{-\\lY\/2 }\n\\endbmatrix.\n\\end{align*}\nTherefore,\n\\begin{align}\\label{eq:trace2length}\nx & =\\; 2 \\sinh\\frac{\\lX}{2} \\\\\ny & =\\;  2 \\sinh\\frac{\\lY}{2} \\notag \\\\\na & =\\; \n-2 \\big(\\sinh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2} \n+ \\cos\\theta  \n\\cosh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\big)\\notag \\\\\nb& =\\;\n-2 \\big(\\sinh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2}\n- \\cos\\theta\n\\cosh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2} \\big). \\notag \n\\end{align}\nThe defining inequalities for these matrices, $x, y > 0$ and $a, b \\le -2$,\ndescribe one component of the set of characters.\nThe \\emph{length identity }\n\\begin{equation}\\label{eq:LengthIdentity}\n\\cosh \\frac{\\lA}{2} \\,+\\, \\cosh\\frac{\\lB}{2}  \\;=\\;  \n2\\ \\sinh\\frac{\\lX}{2}  \\,  \\sinh\\frac{\\lY}{2} .\n\\end{equation}\nresults directly from applying \n\\eqref{eq:trace2length} to \\eqref{eq:trace_identity}.\n\n\n\\begin{thm}\\label{thm:2HXSFrickeSpace}\nThe lengths  of boundary geodesics $\\lA, \\lB \\geq 0$, and the lengths of \norientation-reversing simple closed geodesics  $\\lX,\\lY >0$,\nsubject to \\eqref{eq:LengthIdentity}, \nprovide coordinates for $\\mathfrak{F}(\\topsurf)$. \n\\end{thm}\n\\noindent\nSolve \\eqref{eq:LengthIdentity} for $\\lY$ in terms of the other\nlengths. Thus, $\\mathfrak{F}(\\topsurf)$  identifies with the fibered space whose base\nis the closed first quadrant in $\\mathbb R^2$ ($\\lA, \\lB \\geq 0$) and whose\nfiber $\\mathbb R^+$ ($\\lX >0$) has no boundary.  That is,\n$$\n\\mathfrak{F}(\\topsurf) \\ \\approx \\ \n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+ .\n$$\n\n\\section{Flat Lorentz $3$-manifolds}\\label{sec:3manifolds} \n\nNow we turn from hyperbolic structures on surfaces\nto flat Lorentz structures on $3$-manifolds.\nAfter a brief review of Lorentzian geometry, \nwe prove that every Margulis spacetime is orientable,\neven when the corresponding hyperbolic surface is nonorientable.\n\n\\subsection{Linear Lorentzian geometry}\nLet $\\mathbb{V}$ denote a Lorentzian $3$-dimensional vector space.\nDenote its group of orientation-preserving linear isometries \n${\\o{SO}(2,1)}$.  In keeping with the notation adopted for $\\mathsf{Isom}(\\hp)$, \nwe use uppercase letters to denote elements of this group.\n\nThe nonzero vectors ${\\mathsf v}\\in\\mathbb{V}$ such that \n\\begin{itemize}\n \\item ${\\mathsf v}\\cdot{\\mathsf v} =0$ are called \\emph{lightlike},\n \\item ${\\mathsf v}\\cdot{\\mathsf v} <0$ are called \\emph{timelike}, and\n \\item ${\\mathsf v}\\cdot{\\mathsf v} >0$ are called \\emph{spacelike}.\n\\end{itemize}\nAny lightlike vector lies on one nappe of the \\emph{lightcone}. \nChoose one of these nappes as the  \\emph{future lightcone}.\nAny vector lying on (lightlike)  or inside (timelike)  the future lightcone is said \nto be \\emph{future-pointing}.\n\nThe correspondence between the hyperbolic plane and the set of\ntimelike lines in $\\mathbb{V}$ induces an identification\n$\\mathsf{Isom}(\\hp)\\cong{\\o{SO}(2,1)}$.  \nTransvections \ncorrespond to {\\em hyperbolic} matrices in ${\\o{SO}(2,1)^0}$ with trace greater\nthan 3. The diagonal matrix \\eqref{eq:Transvection} corresponds to \n$$\n\\bmatrix 1 & 0 & 0 \\\\ 0 &  \\cosh(\\ell) & \\sinh(\\ell) \\\\\n0 & \\sinh(\\ell) & \\cosh(\\ell)  \\endbmatrix\n$$\nwith trace $1 + 2\\cosh(\\ell)$.\nParabolic elements of $\\SL(2,\\R)$ correspond to {\\em parabolic\\\/} matrices in \n${\\o{SO}(2,1)^0}$ with trace equal to 3.  The matrix $iP$ of \\eqref{eq:GlideReflection}\nrepresenting a glide-reflection identifies with the diagonal matrix\n$$\n\\bmatrix 1 & 0 & 0 \\\\ 0 &  -\\cosh(\\ell) & -\\sinh(\\ell) \\\\\n0 & -\\sinh(\\ell) & -\\cosh(\\ell)  \\endbmatrix\n$$\nhaving trace $1 -2\\cosh(\\ell) < -1$.\nThe corresponding isometry of $\\Eto$ preserves orientation but reverses time-orientation.\n\nCall $A\\in{\\o{SO}(2,1)}$ {\\em non-elliptic} if it does not fix a timelike\nline.  Equivalently, $A$ corresponds to a glide-reflection, a\ntransvection or a translation of $\\hp$ as above.\n\nSuppose that $A$ is non-elliptic.  Then the 1-eigenspace for $A$ is a\nline spanned by a spacelike or a lightlike vector.  We choose a\nspecific {\\em neutral vector}, denoted $A^0$, by requiring that:\n\\begin{itemize}\n\\item given a timelike vector ${\\mathsf v}$, $\\big( {\\mathsf v}, A^2({\\mathsf v}), A^0 \\big)$ is a right-handed basis for $\\mathbb{V}$.  \n(This determines a unique direction for $A^0$.\nFurthermore this condition is independent of ${\\mathsf v}$.)\n\\item if $\\mathsf{Fix}(A)$ is spacelike, we choose $A^0$ such that $A^0\\cdot A^0=1$.\n\\end{itemize}\nWhen $A$ is hyperbolic (a transvection or glide-reflection), \nits  eigenvalues are $1$, $\\lambda$, and $\\lambda^{-1}$, \nwhere \n$$\n0< \\lambda^2 <1.\n$$\nIn the above example, $\\lambda = \\pm e^{-\\ell}$.\nChoose $A^-$ to be a future-pointing lightlike \\emph{contracting} eigenvector, \nand $A^+$ to be a future-pointing \\emph{expanding} eigenvector: \n$$\nA^2(A^-) = \\lambda^2 A^-,\\qquad  A^2(A^+) = \\lambda^{-2} A^+.\n$$  \nBoth contracting and expanding eigenvectors are lightlike, and the\nneutral vector  $A^0$ is spacelike and well-defined.   \n(See Figure~\\ref{fig:Frames} and compare\n\\cite{CharetteDrumm}.) \n\\begin{figure}[b]\n\\includegraphics[width=8cm]{Frames2.eps\n\\caption{Defining the direction of $A^0$.}\n\\label{fig:Frames}\n\\end{figure}\n\nWhen $A$ is parabolic, $A^0$ may be either future-pointing or\npast-pointing and its Euclidean length is arbitrary.  By convention,\nwe set $A^+=A^-$ to be a future-pointing vector that is parallel to\n$A^0$.\n\n\\subsection{Affine actions}\nLet $\\Eto$ denote the affine space modeled on $\\mathbb{V}$.  The group of {\\em Lorentzian isometries} of $\\Eto$, denoted $\\mathsf{Isom}(\\Eto)$, consists of affine transformations whose linear part preserve the Lorentzian structure on $\\mathbb{V}$.  Equivalently, $\\g\\in\\mathsf{Isom}(\\Eto)$ if and only if its linear part $\\LL(\\g)\\in\\o{O}(2,1)$.\n\nLet $A\\in{\\o{SO}(2,1)}$ be non-elliptic.  \nSince $A$ is linear and therefore fixes the origin, \nthe cyclic group $\\langle A\\rangle$ it generates does not\nact properly on $\\Eto$.\nHowever, for any vector ${\\mathsf u}\\in\\mathbb{V}$ with nonzero projection on $A^0$,\nthe following affine transformation acts properly and freely on\n$\\Eto$:\n$$\ng: p \\longmapsto o+A(p-o) + {\\mathsf u}\n$$\nwhere $o$ is some choice of origin.  The quotient $\\Eto\/\\langle g\\rangle$ is an open solid torus. \n\nMore generally, if $\\Gamma_0\\subset{\\o{SO}(2,1)}$ is a free group, we obtain an affine action with linear part $\\Gamma_0$ by assigning translation parts to a set of free generators.\n\n\\subsection{Margulis spacetimes}\nRecall that a Margulis spacetime is a quotient $M = \\Eto\/\\Gamma$\nwhose fundamental group $\\Gamma$ is free, and acts properly, freely and discretely\nby Lorentz isometries.\n\n\\begin{lemma}\\label{lem:orientable}\nEvery Margulis spacetime with a nonabelian fundamental group is orientable.\n\\end{lemma}\n\\begin{proof}\nFor any Margulis spacetime $M$, \nthe affine holonomy group $\\Gamma$\nacts freely on $\\Eto$.\nAn affine transformation whose linear\npart does not have $1$ as an eigenvalue\nfixes a point in $\\Eto$.\nTherefore, for every $\\gamma\\in\\Gamma$,\nits linear part \n$\\LL(\\gamma)$ must have $1$ as an eigenvalue.\n\nIf $\\LL(\\gamma)\\in\\o{O}(2,1)\\setminus{\\o{SO}(2,1)}$, then one of its eigenvalues \nequals $-1$.   If $1$ is also an eigenvalue, then $A^2 = I$.\nThus, if $\\LL(\\Gamma)\\not\\subset {\\o{SO}(2,1)}$, \nthen some $\\gamma\\in\\Gamma$ will have \ntrivial linear part, and therefore is\na translation. \nBy \\cite{FriedGoldman}, $\\Gamma$ contains\nno translations.\n\\end{proof}\n\\section{Affine deformations and cocycles}\\label{sec:deformations} \nAffine deformations of a two-holed cross-surface $\\surf$\nare parametrized by Margulis invariants of $X,Y,A,B$.  \nWe start with a general\ndiscussion of the space of affine deformations of the linear holonomy\nof a surface and the Margulis invariant.  At the end of the section,\nwe use the Margulis invariant to provide coordinates for the\ndeformation space for the two-holed cross-surface.\n\n\\subsection{Cocycles}\nLet $\\surf$ be an arbitrary hyperbolic surface with linear holonomy $\\Gamma_0=\\rho_0(\\pi_1(\\surf))\\subset{\\o{SO}(2,1)}$.  \nRecall\nthat an affine deformation of $\\Gamma_0$\nis a lift $\\Gamma_0\\xrightarrow{\\rho}\\mathsf{Isom}(\\Eto)$ such that $\\rho_0 = \\LL\\circ \\rho$.\nLifts correspond to cocycles, that is, maps\n$$\n\\Gamma_0  \\stackrel{{\\mathsf u}}\\longrightarrow \\mathbb{V}\n$$\nsatisfying\n$$\n{\\mathsf u}( XY) \\;=\\; {\\mathsf u}(X) + \\rho_0(X) {\\mathsf u}(Y)\n$$\nHere the affine deformation is defined by:\n$$\np \\stackrel{\\rho(X)}\\longmapsto o+\\rho_0(X)(p-o) + {\\mathsf u}(X).\n$$\nDenote the space of cocycles  $\\Gamma_0\\rightarrow\\mathbb{V}$ by $Z^1(\\Gamma_0,\\rto)$.\n\nWhen $\\Gamma_0$ %\nis free, as is the case for a two-holed cross-surface, a cocycle is determined by its values on a free basis.\nFurthermore, the values on a free basis are completely arbitrary.\n\nTwo cocycles determine translationally conjugate affine deformations\nif and only if they are {\\em cohomologous,\\\/} that is, they differ\nby a coboundary\n$$\n{\\mathsf u}(X) := {\\mathsf v} - \\rho_0(X) {\\mathsf v}\n$$\nwhere ${\\mathsf v}\\in\\mathbb{V}$ is the vector effecting the translation.\nThe resulting set of translational conjugacy classes of affine deformations\n(cohomology classes of cocycles)\ncompose the cohomology group $H^1(\\Gamma_0,\\rto)$. \n\nIn the present case, when $\\Gamma_0$ is free of rank two,\nand $\\dim(\\mathbb{V})=3$, the space \n$Z^1(\\Gamma_0,\\rto)$ of cocycles is $6$-dimensional.\nThe space of coboundaries is $3$-dimensional.\nTherefore the cohomology $H^1(\\Gamma_0,\\rto)$ has dimension $3$.\n \n\\subsection{The Margulis invariant}\n\nContinuing with the notation above, the {\\em Margulis invariant\\\/} of the affine deformation $\\rho(X)$ is the neutral projection of the translational part ${\\mathsf u}$:\n$$\n\\oa{{\\mathsf u}}{X} \\;:=\\; {\\mathsf u} \\cdot X^0\n$$\nThe Margulis invariant \nis everywhere nonzero if and only if the affine deformation\nis free.  Moreover if ${\\mathsf u},{\\mathsf v}$ are cohomologous then $\\oa{{\\mathsf u}}{X}=\\oa{{\\mathsf v}}{X}$.  Furthermore, fixing the cocycle ${\\mathsf u}$, the Margulis invariant is a class function on $\\Gamma_0$.  \n\nThe following basic fact\n(Margulis's Opposite Sign Lemma) \nwas proved in \\cite{Margulis1,Margulis2} \nfor hyperbolic elements and extended in\n\\cite{CharetteDrumm} to groups with parabolic elements. \nThe survey article~\\cite{Abels} containsa lucid description of the ideas in \nMargulis's original  proof. \n\\begin{lemma}\\label{lemma:oppsign}\nLet $g, h\\in\\mathsf{Isom}(\\Eto)$ be non-elliptic.  \nIf the Margulis invariants for $g$ and $h$ have \nopposite signs then $\\langle g , h \\rangle$ does not act properly. \n\\end{lemma}\n\n\n\\subsection{Deformations of hyperbolic structures}\n\nAffine deformations also correspond to \n{\\em infinitesimal deformations\\\/}\nof the marked hyperbolic surface $\\Sigma \\to S$\n\n\\cite{Goldman_MargulisInvariant,GoldmanMargulis}).\nConsider a smooth family $S_t$ of hyperbolic surfaces, with \nsmoothly varying markings \n$$\n\\Sigma \\xrightarrow{m_t} S_t\n$$ \nand holonomy representation \n$\\pi_1(\\Sigma)\\xrightarrow{\\rho_t}\\SL(2,\\C)$.  \nLet ${\\mathsf u}$ be the cocycle tangent to the space of representations\nat $t=0$ .\nBy \\cite{GoldmanMargulis},\nthe Margulis invariant identifies with the derivative of the {\\em geodesic length function $\\ell_\\gamma$,\\\/}\nwhere $\\gamma\\in\\fg$. Specifically, let $\\ell_{\\gamma(t)}\\in\\mathbb R_+$ denote the length of\nthe closed geodesic in $S_t$ in the free homotopy class determined by $(m_t)_*(\\gamma)\\in\\pi_1(S)$.\nThen:\n$$\n\\oa{{\\mathsf u}}{\\rho_0(\\gamma)} \\;=\\;  \\frac{d \\ell_{\\gamma(t)} }{dt} .\n$$\nFurthermore: \n\\begin{equation}\\label{eq:Lengths2Traces}\n\\mathsf{tr}\\big( \\rho_t(\\gamma)\\big) \\;=\\; \\pm 2 \n\\begin{cases}\n\\cosh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~preserves orientation} \\\\\ni \\sinh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~reverses orientation} \n\\end{cases}\n\\end{equation}\nDifferentiating  \\eqref{eq:Lengths2Traces} implies:\n\\begin{equation}\\label{eq:dLengths2Traces}\n\\frac{d}{dt} \\mathsf{tr} \\big( \\rho_t(\\gamma) \\big)   \\;=\\; \\pm\n\\begin{cases}\n\\alpha_\\gamma \\sinh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~preserves orientation} \\\\\ni\\alpha_\\gamma \\cosh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~reverses orientation} \n\\end{cases}\n\\end{equation}\n\n\n\\subsection{Coordinates for the four-sided deformation space}\n\\label{sec:coordinates}\n\nFixing $A\\in\\Gamma_0$, the Margulis invariant is a\nlinear functional on the space of cocycles $Z^1(\\Gamma_0,\\rto)$, descending to a\nwell-defined functional on $H^1(\\Gamma_0,\\rto)$.  To reflect this, we modify our\nnotation for the Margulis invariant as in~\\cite{CDG}:\n\\begin{align*}\nH^1(\\Gamma_0,\\rto) & \\xrightarrow{\\tilde{\\alpha}_A} \\mathbb R \\\\\n[{\\mathsf u}] & \\longmapsto \\oa{{\\mathsf u}}{A}\n\\end{align*}\n\nNow assume $\\surf$ is a two-holed cross-surface, with elements $A,B,X,Y\\in\\pi_1(\\surf)$ as in \\S\\ref{sec:thx}.  The invariants of elements $A,X,\\bar{X} A=Y$ determine an isomorphism of vector spaces:  \n\\begin{align}\\label{eq:CDM-coordinates}\nH^1(\\Gamma_0,\\rto) &\\xrightarrow{\\tilde{\\alpha}} \\mathbb{R}^3  \\\\\n [u]  & \\longmapsto\n\\bmatrix \\tilde{\\alpha}_A(u) \\\\ \\tilde{\\alpha}_X(u) \\\\\\tilde{\\alpha}_Y(u) \n\\endbmatrix  \\notag\n\\end{align}\n\nConsider a proper affine deformation.  Lemma~\\ref{lemma:oppsign}\nimplies the Margulis invariants of all nontrivial elements have the\nsame sign.  Namely, consider an infinitesimal deformation of the\nhyperbolic structure on $\\surf$ such that every closed geodesic \n{\\em infinitesimally lengthens} or an infinitesimal deformation where\nevery closed geodesic \\emph{infinitesimally shortens}. \n\nDifferentiating \\eqref{eq:trace_identity} yields:\n\\begin{equation}\\label{eq:deriv_trace_identity}\nda \\, +\\, db \\,+\\, y\\ dx \\ \\,+\\, \\ x\\ dy \\;=\\; 0.\n\\end{equation}\nExpress these quantities using \\eqref{eq:trace2length}, \nits derivatives\n\\eqref{eq:dLengths2Traces},  and use the fact that  \nthe Margulis invariants $\\tilde{\\alpha}_A,\\tilde{\\alpha}_B,\\tilde{\\alpha}_X,\\tilde{\\alpha}_Y$\nof $A,B,X,Y$, are\nthe derivatives $d\\lA, d\\lB, d\\lX, d\\lY$ \nrespectively \\cite{GoldmanMargulis,CharetteDrumm}.\nThe differentiated trace identity \n\\eqref{eq:deriv_trace_identity} then implies:\n\n\n\\begin{align}\\label{eq:alphaCDMrelations}\n&\\Big(\\sinh\\frac{\\lA}{2}\\Big)   \\tilde{\\alpha}_A +  \\Big(\\sinh\\frac{\\lB}{2}\\Big) \\tilde{\\alpha}_B = \\\\\n &  \\qquad\\qquad  \\Big(2\\cosh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2}\\Big) \\tilde{\\alpha}_X \\ +\\ \n\\Big(2 \\sinh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\  \\Big) \n\\tilde{\\alpha}_Y. \\notag  \n\\end{align}\nAfter eliminating $\\tilde{\\alpha}_B$ using \\eqref{eq:alphaCDMrelations}, \nthe positivity of $\\tilde{\\alpha}_A$ and\n$\\tilde{\\alpha}_B$ is written as follows:\n\\begin{align}\\label{eq:muAmuXmuY}\n0 \\   & <  \\tilde{\\alpha}_A  \\\\ \n&  < \n  \\bigg( 2\\opn{csch}\\frac{\\lA}{2} \\cosh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2} \\bigg)\n\\ \\tilde{\\alpha}_X  \\notag \\\\ \n& \\qquad + \n\\bigg( 2\\opn{csch}\\frac{\\lA}{2} \\sinh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\bigg)\\ \n\\tilde{\\alpha}_Y \\notag\n\\end{align}\nRecall that $\\Dd$ denotes the set of deformations where the\nfunctionals $\\tilde{\\alpha}_X , \\tilde{\\alpha}_Y , \\tilde{\\alpha}_A , \\tilde{\\alpha}_B $ all have the same sign.  Let $\\Dd_+\\subset\\Dd$ denote those whose signs are all positive.  Then the set of lengthening deformations lies inside the\nset $\\Dd_+$\nand is defined in terms of the linear coordinates $\\tilde{\\alpha}_A,\\tilde{\\alpha}_X,\\tilde{\\alpha}_Y$\nby \\eqref{eq:muAmuXmuY} and the conditions $\\tilde{\\alpha}_X > 0$ and $\\tilde{\\alpha}_Y > 0$.\n\n\n$\\Dd_+$ is a four-sided cone, and is thus spanned by four rays.\nEach of these rays can be defined by\nthe linear equation \\eqref{eq:alphaCDMrelations} together\nwith one of the four conditions:\n\\begin{align*}\n\\tilde{\\alpha}_Y, \\tilde{\\alpha}_B > 0,\\; &\\tilde{\\alpha}_X = \\tilde{\\alpha}_A = 0 \\\\\n\\tilde{\\alpha}_X, \\tilde{\\alpha}_B > 0,\\; &\\tilde{\\alpha}_Y = \\tilde{\\alpha}_A = 0 \\\\\n\\tilde{\\alpha}_X, \\tilde{\\alpha}_A > 0,\\; &\\tilde{\\alpha}_Y = \\tilde{\\alpha}_B = 0 \\\\\n\\tilde{\\alpha}_Y, \\tilde{\\alpha}_A > 0,\\; &\\tilde{\\alpha}_X = \\tilde{\\alpha}_B = 0.\n\\end{align*}\n\\noindent \nThe set of shortening deformations lies inside $\\Dd_-$, where \n$$\n\\Dd_- = -\\Dd_+. \n$$ \n\n\\begin{figure}[bh]\n\\includegraphics[scale=2.3]{Quadrilateral.eps}\n\\caption{The four-sided deformation space\nfor a two-holed cross-surface}\n\\label{fig:DeformationSpace}\n\\end{figure}\n\n\\section{Crooked fundamental domains}\n\nIn this final section we prove the Main Theorem stated in the\nIntroduction.  Explicitly, we show that every proper affine\ndeformation admits a crooked fundamental domain bounded by four\ndisjoint crooked planes.  It follows that $\\Eto\/\\Gamma$ is a solid\nhandlebody of genus two.\n\nAs in the case of the three-holed sphere~\\cite{CDG}, \nit suffices\nto consider all affine deformations arising from a single\nconfiguration of crooked planes. This configuration is modeled\non the fundamental domain for $S$ developed in \n\\S\\ref{sec:funddomainsHypSurf}. \nThese crooked plane configurations correspond to \ndecompositions of $S$ into two ideal triangles  bounded by three different geodesics. Some of these crooked\nplane configurations also describe the entire space of proper affine deformations. \nThe more complicated\ncase of the one-holed torus requires the use of multiple configurations\nof crooked planes to cover all the proper affine deformations~\\cite{CDGp}.\nThat is why we obtain a finite-sided\ndeformation space, in contrast with the case of the one-holed torus,\nfor example.\n\nWe provide definitions and some background on crooked planes in the Appendix.\n\n\\subsection{Configuring crooked planes}\\label{sec:CrookedFD}\n\nGiven a spacelike vector ${\\mathsf v}\\in\\mathbb{V}$ and $p\\in\\Eto$, let ${\\mathcal C}({\\mathsf v},p)$\ndenote the crooked plane with direction vector ${\\mathsf v}$ and vertex $p$.\nIts complement in $\\Eto$ is a pair of crooked halfspaces.\n\nAs one-dimensional spacelike subspaces of $\\mathbb{V}$ bijectively\ncorrespond to hyperbolic lines in ${\\mathsf H}^2$, so do parallelism classes of\ncrooked planes.  Thus to any configuration of lines in ${\\mathsf H}^2$ we can\nassociate a corresponding configuration of crooked planes, up to\nchoice of vertices.  In particular, every ideal triangle in $\\hp$\ncorresponds to a triple of crooked planes such that every pair of\ndirection vectors shares a common orthogonal lightlike line.  Moreover\nsuch a triple determines a triple of pairwise disjoint crooked\nhalfspaces, for appropriate choices of vertices.\n\nCrooked planes enjoy the following useful property:\norientation-preserving Lorentzian isometries map crooked planes to\ncrooked planes.  Explicitly, if $g\\in{\\o{SO}(2,1)}$ then:\n$$\ng\\big({\\mathcal C}({\\mathsf v},p)\\big)={\\mathcal C}\\big(\\LL(g)({\\mathsf v}),g(p)\\big).$$ \nMoreover, if these crooked\nplanes are disjoint, then there exists a crooked halfspace $\\mathcal{H}$ in the\ncomplement of ${\\mathcal C}({\\mathsf v},p)$ such that the closures of $\\mathcal{H}$ and $g(\\mathcal{H})$\nare disjoint.  Thus crooked planes are suitable for Klein-Maskit\ncombination arguments, {\\em even for groups containing  glide-reflections}.\n\n\\subsection{Disjointness}\n\nStart with four crooked planes all with the same vertex and\ncorresponding to the ideal quadrilateral $\\mathcal Q$ pictured in\nFigure~\\ref{fig:IdealQuadrilateral}.  We  associate vertices to\neach crooked plane which make them disjoint.  This assignment of\nvertices yields a proper affine deformation of $\\Gamma_0=\\langle\nA,X\\rangle$.\n\nWe introduce \na convention\non \nlightlike vectors.  Recall that glide-reflections in $\\mathsf{Isom}(\\hp)$\nidentify with isometries in ${\\o{SO}(2,1)}$ which interchange the future and\npast null cones.  To facilitate calculations, take all lightlike vectors\nto be future-pointing. If $X$ corresponds to a\nglide-reflection and ${\\mathsf v}$ is a future-pointing vector then write all\nexpressions involving the action of $X$ on ${\\mathsf v}$ in terms of $X(-{\\mathsf v})$\nwhich is again future-pointing.\n\nAs in~\\cite{CDG}, it suffices to consider a triple of crooked planes\ncorresponding to one of the ideal triangles forming $\\mathcal Q$.  Let\n${\\mathsf v}_0$, ${\\mathsf v}_A$, ${\\mathsf v}_X$ be a triple of consistently oriented\nunit-spacelike vectors such that:\n\\begin{align*}\n{\\mathsf v}_0^\\perp & = \\langle\\xm{A},X(-\\xp{B})\\rangle \\\\\n{\\mathsf v}_A^\\perp & = \\langle A(\\xp{B}),\\xm{A}\\rangle \\\\\n{\\mathsf v}_X^\\perp & = \\langle X(-\\xp{B}),A(\\xp{B}) \\rangle.\n\\end{align*}\nThese will be the directing vectors of our triple of crooked planes.  \n\nGiven $(p_0,p_A,p_X)\\in \\Eto \\times \\Eto \\times \\Eto$, \ndefine the cocycle \n${\\mathsf u}_{(p_0,p_A,p_X)} \\;\\in\\; Z^1(\\Gamma_0,\\rto)\n$ \nas follows:\n\n\\begin{align*}\n{\\mathsf u}_{(p_0,p_A,p_X)}(A) &:=  p_A - p_0  \\\\\n{\\mathsf u}_{(p_0,p_A,p_X)}(X) &:=  p_X - p_0.\n\\end{align*}\nSince $A, X$ freely generate $\\fg$, \nthese conditions uniquely determine the cocycle ${\\mathsf u}_{(p_0,p_A,p_X)}$.\n\n\nThis cocycle corresponds to the following triple of crooked planes: \n\\begin{align*}\n{\\mathcal C}_0 & := \\;\n{\\mathcal C}({\\mathsf v}_0 ,p_0) \\\\\n{\\mathcal C}_A & := \\;\n{\\mathcal C}({\\mathsf v}_A , p_A) \\\\\n{\\mathcal C}_X & := \\;\n{\\mathcal C}({\\mathsf v}_X,  p_X).\n\\end{align*}\nIf $\\rho$ is the affine deformation corresponding to\n${\\mathsf u}_{(p_0,p_A,p_X)}$, then ${\\mathcal C}_0$ bounds a crooked halfspace whose\nclosure contains $\\rho(A)^{-1}({\\mathcal C}_A)$ and $\\rho(X)^{-1}({\\mathcal C}_X)$.\nNote that the quadruple\n$$ \n{\\mathcal C}_A,{\\mathcal C}_X,\\rho(A)^{-1}({\\mathcal C}_A),\\rho(X)^{-1}({\\mathcal C}_X) \n$$ \ncorresponds to $\\mathcal Q$.\n\nWe would like these crooked planes to be disjoint.  \nTo this end, consider the three quadrants:\n\\begin{align*}\n\\mathscr Q_0 & :=\\; \\Quad{\\xm{A}, -X(-\\xp{B})}  \\\\\n\\mathscr Q_A & :=\\; \\Quad{A(\\xp{B}), -\\xm{A}} \\\\\n\\mathscr Q_X & :=\\; \\Quad{X(-\\xp{B}),- A(\\xp{B})} \\\\\n\\end{align*}\nBy Lemma~\\ref{lem:disjointasym}, the triple of crooked planes ${\\mathcal C}_0,{\\mathcal C}_A ,{\\mathcal C}_X$ are pairwise disjoint as long as $(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$.\n\n\\begin{thm}\\label{thm:tame}\nLet \n$$\n(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X.\n$$\nThen the affine deformation defined by\n$$\n{\\mathsf u}_{(p_0,p_A,p_X)} \\;\\in\\; Z^1(\\Gamma_0,\\rto)\n$$\nis tame.\n\\end{thm}\n\n\\begin{proof}\nApply the Kissing Lemma (Lemma~\\ref{lem:kissing1}) to the crooked planes \nin the halfspace bounded by ${\\mathcal C}({\\mathsf v}_0,p_0)$, to obtain pairwise disjoint crooked planes.  By Theorem~\\ref{thm:disjointCPs}, they bound a fundamental domain.\n\\end{proof}\n\nFigure~\\ref{fig:FourwTwoKiss} shows a quadruple of crooked planes, to which we apply the Kissing Lemma to obtain pairwise disjoint crooked planes. \n\n\\begin{figure}\n\\includegraphics[width=6.3in]{FourwTwoKiss.eps} \n\\caption{Four crooked planes.  The two on the right share a wing.}\n\\label{fig:FourwTwoKiss} \n\\end{figure}\n\n\\subsection{Proper deformations are tame}\nNow we conclude the proof.  Recall that all tame deformations are\nproper.  The Margulis invariants  of the four\nelements $X,Y,A,B$ are all of the same sign for every proper\ndeformation.  That is,\n$$\n\\Ta \\; \\subset \\; \\Pr \\; \\subset \\; \\Dd .\n$$\nWe now show that $\\Dd \\; \\subset \\; \\Ta$ .\n\nAs before, Lemma~\\ref{lemma:oppsign} implies that each of these \nsets has two connected components which are opposites of each other. \nFor proper deformations, \n$$\n\\Pr= \\Pr_+\\cup \\Pr_-\n$$ \nand $\\Pr_-= - \\Pr_+$. \nFor tame deformations, \n$$\n\\Ta= \\Ta_+ \\cup \\Ta_-\n$$ \nand $\\Ta_- = - \\Ta_+$. \nThus \n\\begin{align*}\n\\Ta_+   & \\subset \\Pr_+  \\subset \\Dd_+ \\\\\n\\Ta_-   & \\subset \\Pr_-  \\subset \\Dd_- \n\\end{align*}\nTheorem~\\ref{thm:tame} asserts that the image of the mapping\n\\begin{align*}\n\\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X &\\longrightarrow H^1(\\Gamma_0,\\rto) \\\\\n(p_0,p_A,p_X) &\\longmapsto [{\\mathsf u}_{(p_0,p_A,p_X)}]\n\\end{align*}\nlies in $\\Ta$.\n\n\nIn fact we shall show that this image is exactly\n$\\Dd_+$. The proof for the positive components\nimplies\nthe same statements for the negative components.\n\nWe will in fact consider cohomology classes in the closure of $\\Ta_+$.  Write:\n\\begin{equation}\\label{eq:vertexdef}\n\\begin{array}{rcl}\np_0 &:= &r_0 \\xm{A} - s_0 X(-\\xp{B}) \\\\\np_A &:= &r_A A(\\xp{B}) - s_A \\xm{A}  \\\\\np_X &:=& r_X X(-\\xp{B}) - s_X A(\\xp{B}) \n\\end{array}\n\\end{equation}\nwhere \n$$\nr_0, s_0, r_A, s_A, r_X, s_X \\geq 0.\n$$\nAssigning a zero value to any of these coefficients yields a configuration of \n{\\em kissing\\\/} crooked planes, any two of  which intersect in a point, \na ray or a halfplane.\n\nApplying the definition of a cocycle:\n\\begin{lemma}\\label{lem:yb}\nFor any cocycle ${\\mathsf u}$,\n\\begin{itemize}\n\\item ${\\mathsf u}(Y) \\;=\\; \\bar{X} \\big({\\mathsf u}(A) - {\\mathsf u}(X)\\big)$;\n\\item ${\\mathsf u}(B) \\;=\\; \\bar{Y} \\big({\\mathsf u}(X) - {\\mathsf u}(Y)\\big)$.\n\\end{itemize}\\end{lemma}\n\\noindent\nConsequently:\n\\begin{equation}\\label{eq:udef} \n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  p_A - p_0 \\\\\n{\\mathsf u}(X)\t & = &  p_X - p_0  \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(p_A - p_X) \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(p_X - p_0)- \\bar{A}( p_A - p_X )  \n\\end{array}\n\\end{equation}\nTherefore:\n\\begin{equation}\\label{eq:margulisoftriple}\n\\tilde{\\alpha}([{\\mathsf u}_{(p_0,p_A,p_X)}])=\n\\begin{bmatrix}\n(r_A\\xp{B}+s_0X(-\\xp{B}))\\cdot A^0 \\\\\n(-(r_X+s_0)\\xp{B}-s_XA(\\xp{B})-r_0\\xm{A})\\cdot X^0\\\\\n((r_A+r_X+s_X)\\xp{B}-s_AX^{-1}(\\xm{A}))\\cdot Y^0\n\\end{bmatrix}\n\\end{equation}\n\n\\noindent\nThe following lemma is also immediate:\n\\begin{lemma}\\label{lem:alphasXYA}\nLet $(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$ be as above.\n\\begin{itemize}\n\\item \n${\\mathsf u}_{(p_0,0,0)}(Y) \\;=\\; 0$ so \n$\\tilde{\\alpha}_Y({\\mathsf u}_{(p_0,0,0)}) \\;=\\; 0$.\n\\item \n${\\mathsf u}_{(0,p_A,0)}(X) \\;=\\; 0$ so \n$\\tilde{\\alpha}_X({\\mathsf u}_{(0,p_A,0)}) \\;=\\; 0$.\n\\item \n${\\mathsf u}_{(0,0,p_X)}(A) \\;=\\; 0$ so \n$\\tilde{\\alpha}_A({\\mathsf u}_{(0,0,p_X)}) \\;=\\; 0$.\n\\end{itemize}\n\\end{lemma}\n\nThe deformation space $\\Dd_+$ is the (open) convex hull of four rays.\nWe will exhibit a cocycle on each ray and show that its $\\mu$-coordinates are all nonnegative. \n\nFigure~\\ref{fig:FuturePointing} serves as a visual aid in the\ncalculation of $\\mu$-coordinates, by showing where the relevant\nfuture-pointing vectors are located relative to the axes of $X$ and $Y$. \n\n\\begin{figure}[h]\n\\psfrag{m}{$X^-$}\n\\psfrag{V}{$X^+$}\n\\psfrag{Z}{$Y^-$}\n\\psfrag{W}{$Y^+$}\n\\psfrag{X}{$X$}\n\\psfrag{Y}{$Y$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$B$}\n\\psfrag{P}{$A^Y\\,=\\,A^{\\bar{X}}$\n\\psfrag{Q}{$B^Y\\,=\\,B^X$\n\\psfrag{o}{$\\mathsf{dev}(\\tbasepoint)$}\n\\includegraphics[scale=0.6]\n{TheGeometry.eps}\n\\caption{Computing cocycles on the edge of $\\Dd_+$. The locations of relevant future-pointing vectors are shown.  The figure depicts the case when $A,B$ are hyperbolic; in the case where either isometry is parabolic, the corresponding invariant axis shrinks to a point.}\n\\label{fig:FuturePointing}\n\\end{figure}\n\n\n\\bigskip\n\\noindent{{\\bf First ray:~}$\\tilde{\\alpha}_B^{-1}(0)\\cap\\tilde{\\alpha}_Y^{-1}(0)$}\n\n\\smallskip\n\\noindent\nFor $\\tilde{\\alpha}_B =\\tilde{\\alpha}_Y  =  0$, \nlet $(p_0,p_A,p_X)  =  (-s_0 X(-\\xp{B}), 0, 0)$ with $s_0>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  s_0 X(-\\xp{B}) \\\\\n{\\mathsf u}(X)\t & = &  s_0 X(-\\xp{B}) \\\\\n{\\mathsf u}(Y) \t & = & \\mathsf{0} \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(s_0 X(-\\xp{B})) = -s_0\\xp{B}\n\\end{array}\n\\end{equation*}\n\n\\noindent\nBecause $\\xp{B}\\cdot \\xo{B} = 0$, we know that\n$\\tilde{\\alpha}_Y({\\mathsf u})=\\tilde{\\alpha}_B({\\mathsf u})=0$. Recall that $X$ is a glide\nreflection, so that the future pointing vector $X(-\\xp{B})$ lies\nabove  \nthe axis of $X$ in\nFigure~\\ref{fig:FuturePointing}. Therefore, $X(-\\xp{B}) \\cdot\n\\xo{X}>0$ and $X(-\\xp{B}) \\cdot \\xo{A}>0 $, so that\n$$\n\\tilde{\\alpha}(A) =s_0 X(-\\xp{B}) \\cdot  \\xo{A} >0\n$$\nand\n$$\n\\tilde{\\alpha}(X) =s_0 X(-\\xp{B}) \\cdot  \\xo{X} >0.\n$$\n\n\\bigskip\n\\noindent{{\\bf Second ray:~}$\\tilde{\\alpha}_B^{-1}(0)\\cap\\tilde{\\alpha}_X^{-1}(0)$}\n\n\\smallskip\n\n\\noindent\nFor $\\tilde{\\alpha}_B = \\tilde{\\alpha}_X  =  0$, let \n$(p_0,p_A,p_X)  =  (0, r_A A(\\xp{B}), 0)$ with $r_A >0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  r_A A(\\xp{B}) \\\\\n{\\mathsf u}(X)\t & = &  \\mathsf{0} \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(r_A A(\\xp{B})) =  - r_A Y(-\\xp{B}) \\\\\n{\\mathsf u}(B)\t & = &  -\\bar{A}(r_A A(\\xp{B})) = -r_A \\xp{B}\n\\end{array}\n\\end{equation*}\nClearly  $\\tilde{\\alpha}_X({\\mathsf u})=\\tilde{\\alpha}_B({\\mathsf u})=0$. The future pointing \nvector $Y(-\\xp{B})$ lies to the right of the axis of $Y$ in \nFigure~\\ref{fig:FuturePointing},  so that\n$Y(-\\xp{B}) \\cdot  \\xo{Y}<0$. Because $\\xp{B} \\cdot  \\xo{A}>0$, then $A(\\xp{B}) \\cdot  \\xo{A}>0$.\nThus, \n$$\n\\tilde{\\alpha}(A) =r_A A(\\xp{B}) \\cdot  \\xo{A} >0 \n$$\nand\n$$\n\\tilde{\\alpha}(Y) =- r_A Y(-\\xp{B})  \\cdot  \\xo{Y} >0 .\n$$\n\n\\bigskip\n\\noindent{{\\bf Third ray:~}$\\tilde{\\alpha}_A^{-1}(0)\\cap\\tilde{\\alpha}_X^{-1}(0)$\n\n\\smallskip\n\\noindent\nFor $\\tilde{\\alpha}_A =\\tilde{\\alpha}_X  =  0$, let \n$(p_0,p_A,p_X)  =  (0, -s_A \\xp{A}, 0)$ with $s_A>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  -s_A \\xp{A}\\\\\n{\\mathsf u}(X)\t & = &  \\mathsf{0} \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(-s_A \\xp{A})\\ =\\  s_A \\bar{X}(-\\xp{A}) \\\\\n{\\mathsf u}(B)\t & = &  -\\bar{A}(-s_A \\xp{A})\\ =\\  s_A \\xp{A}\n\\end{array}\n\\end{equation*} \nIt is clear that $\\tilde{\\alpha}_A({\\mathsf u})=\\tilde{\\alpha}_X({\\mathsf u})=0$. Because $\\xp{A}$ is a fixed point\nof $A = XY$, the future pointing vector \n$\\bar{X}(-\\xp{A})=Y(-\\xp{A})$ lies to the left of the axis for $Y$ so that\n$\\bar{X}(-\\xp{A}) \\cdot  \\xo{Y}>0$. It is also evident that $\\xp{A} \\cdot  \\xo{B}>0$.\nThus\n$$\n\\tilde{\\alpha}(Y) =s_A \\bar{X}(-\\xp{A})  \\cdot  \\xo{Y} >0\n$$\nand\n$$\n\\tilde{\\alpha}(B) =- s_A \\xp{A}  \\cdot  \\xo{B} >0 \n$$\n\n\\newpage\n\\noindent{{\\bf Fourth ray:~}$\\tilde{\\alpha}_A^{-1}(0)\\cap\\tilde{\\alpha}_Y^{-1}(0)$}\n\n\\smallskip\n\n\\noindent\nFor $\\tilde{\\alpha}_A\\;=\\;\\tilde{\\alpha}_Y \\; = \\; 0$, let  \n$(p_0,p_A,p_X) \\; = \\; (r_0 \\xm{A}, 0, 0)$ with $r_0>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  -r_0 \\xm{A}\\\\\n{\\mathsf u}(X)\t & = &  -r_0 \\xm{A} \\\\\n{\\mathsf u}(Y) \t & = & \\mathsf{0} \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(-r_0 \\xm{A})\\ =\\  r_0\\bar{Y}(-\\xm{A}) .\n\\end{array}\n\\end{equation*}\nCertainly $\\tilde{\\alpha}_A({\\mathsf u})=\\tilde{\\alpha}_Y({\\mathsf u})=0$. It is also clear that \n$\\xm{A} \\cdot  \\xo{X}<0$.  \n\nIn order to locate $\\bar{Y}(-\\xm{A})$, observe that $\\xm{A}$ is inside the\ncircular arc from $X(-\\xp{B})$ clockwise to  $\\xm{Y}$ \n(see Figure~\\ref{fig:IdealQuadrilateral}). \nBecause $B=\\bar{Y}X$,  $X(-\\xp{B}) = Y(-\\xp{B})$ and $Y$ maps the arc from\n$X(-\\xp{B})$ clockwise to  $\\xm{Y}$  to the arc from $\\xm{Y}$ to $\\xp{B}$. \nIn particular, $\\bar{Y}(-\\xm{A}) \\cdot  \\xo{B}>0$ \nso that: \n$$\n\\tilde{\\alpha}(X) =-r_0 \\xm{A}  \\cdot  \\xo{X} >0\n$$\nand\n$$\n\\tilde{\\alpha}(B) =r_0\\bar{Y}(-\\xm{A}) \\cdot  \\xo{B} >0 \n$$\n\nTherefore, every cohomology class in the interior of the convex hull of these four rays is represented by a cocycle ${\\mathsf u}_{(p_0,p_A,0)}$ such that \n$$\nr_0, s_0, r_A, s_A>0.\n$$  \nThis only yields a configuration of crooked planes where ${\\mathcal C}_0$ and ${\\mathcal C}_A$ are disjoint, but each kisses ${\\mathcal C}_X$.  However, \nEquation~\\eqref{eq:margulisoftriple} implies that by slightly perturbing $s_A$ and $r_0$, we can make $r_X$ and $s_X$ both strictly positive within the same cohomology class.  Finally, since the new values $(p'_0,p'_A,p_X)$ are in $\\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$,  $[{\\mathsf u}_{(p_0,p_A,0)}]\\in\\Ta_+$ and thus $\\Dd_+\\subset\\Ta_+$ as claimed.\n\\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\input{Introduction}\n\n\\section{System Model}\n\\input{SystemModel}\n\\subsection{Energy Transfer Phase}\n\\input{DownlinkStage}\n\\subsection{Information Transmission Phase}\n\\input{UplinkStage}\n\n\\section{Problem Formulation and Optimal Solution}\n\\label{Section:ProblemFormulationAndOptimalSolution}\n\\input{ProblemFormulationSolution}\n\\subsection{Problem Formulation}\n\\input{ProblemFormulation}\n\\subsection{Single-User WPCNs}\n\\label{Section:SingleUser}\n\\input{SingleUserSystems}\n\n\\subsection{Multi-User WPCNs}\n\\input{MultiUserOptimal}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:FunctionPsi})}\n\t\\label{Section:OptimalFunctionPsi}\n\t\\input{FunctionSolution}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:OptimalResourceAlloc})}\n\t\\label{Section:OptimalResourceAllocation}\n\t\\input{ResourceAllocSolution}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:OptimalTransmitVectors})}\n\t\\label{Section:OptimalVectors}\n\t\\input{VectorsSolution}\n\n\\section{Low-Complexity Design of Multi-User WPCNs}\n\\input{MultiUserSuboptimal}\n\\subsection{Massive MISO WPCNs}\n\\label{Section:MassiveMIMO}\n\\input{MassiveMIMOScheme}\n\\subsection{Suboptimal MRT-based Scheme}\n\\input{ScaledMassiveMIMO}\n\\subsection{Suboptimal SDR-based Scheme}\n\\input{SuboptimalScheme3}\n\n\\section{Numerical Results}\n\\label{Section:SimulationResults}\n\\input{NumericalResults}\n\\subsection{Simulation Setup}\n\\input{SimulationSetup}\n\\subsection{Complexity Analysis}\n\\input{ComplexityAnalysis}\n\\subsection{Performance Analysis}\n\\input{PerformanceAnalysis}\n\n\\section{Conclusions}\n\\input{Conclusions}\n\n\\appendices\n\n\t\\renewcommand{\\thesection}{\\Alph{section}}\n\t\\renewcommand{\\thesubsection}{\\thesection.\\arabic{subsection}}\n\t\\renewcommand{\\thesectiondis}[2]{\\Alph{section}:}\n\t\\renewcommand{\\thesubsectiondis}{\\thesection.\\arabic{subsection}:}\t\n\t\\section{Proof of Proposition \\ref{Theorem:SingleUser}}\n\t\\label{Appendix:PropSU}\n\t\\input{ProofPropSU}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp1}}\n\t\\label{Appendix:Prop1}\n\t\\input{ProofProp1}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp2}}\n\t\\label{Appendix:Prop2}\n\t\\input{ProofProp2}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp3}}\n\t\\label{Appendix:Prop3}\n\t\\input{ProofProp3}\n\t\\section{Proof of Lemma \\ref{Theorem:Lemma}}\n\t\\label{Appendix:LemmaProof}\n\t\\input{ProofLemma}\n\t\\section{Proof of Proposition \\ref{Theorem:MassiveMIMO}}\n\t\\label{Appendix:Prop5}\n\t\\input{ProofProp5}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSystems of coupled bulk--surface partial differential equations arise in many engineering and natural science applications. Examples include multiphase fluid dynamics with soluble or insoluble surfactants~\\cite{GrossReuskenBook}, dynamics of biomembranes~\\cite{bonito2011dynamics}, crystal growth~\\cite{Crystal},  signaling in biological networks~\\cite{Cells}, and transport of solute in fractured porous media \\cite{alboin2002modeling}. In these and other applications, partial differential equations defined in a volume domain are coupled to another PDEs posed on a surface. The surface may be embedded in the bulk or belong to a boundary of the volume domain.\n\nRecently, there has been a growing interest in developing methods for the numerical treatment of bulk--surface coupled PDEs.\nDifferent approaches  can be distinguished depending on how the surface is recovered and equations are treated.\nIf a  tessellation of the volume into tetrahedra is available that fits the surface, then it is natural to introduce finite element spaces in the volume and on the induced triangulation of the surface. The resulting \\textit{fitted} bulk--surface finite element method was studied  for the stationary bulk--surface advection--diffusion equations \\cite{ER2013}, for non-linear reaction--diffusion systems modelling biological pattern formation  \\cite{madzvamuse2016bulk,madzvamuse2015stability}, for the equations of the two--phase flow with surfactants~\\cite{barrett2015stable,barrett2015stable2},  Darcy and transport--diffusion  equations in fractured porous media~\\cite{alboin2002modeling}. \n\n\\textit{Unfitted} finite element methods allow the surface to cut through the background tetrahedral mesh. In the class of finite element methods also known as cutFEM, Nitsche-XFEM or TraceFEM, standard background finite element spaces are employed, while the integration is performed over cut domains and over the embedded surface~ \\cite{cutFEM,TraceFEM}. Additional stabilization terms are often added to ensure the robustness of the method with respect to small cut elements. The advantages of the unfitted approach are the efficiency in handling implicitly defined surfaces, complex geometries,   and  the flexibility in dealing with evolving domains. In the context of bulk--surface coupled problems,  cut finite element methods were recently applied to treat  stationary bulk--surface advection--diffusion equations \\cite{gross2015trace},\ncoupled bulk-surface problems on time-dependent domains \\cite{hansbo2016cut}, coupled elasticity problems \\cite{cenanovic2015cut}. The hybrid  method developed in this paper belongs to the general class of unfitted methods and resembles the TraceFEM in how the surface PDE is treated.\n\nThe methods discussed above treat surfaces and interfaces sharply, i.e. as lower-dimensional manifolds. In the present paper we also  consider sharp interfaces. For the application of phase--field or other diffuse-interface approaches for coupled bulk--surface PDEs see,  for example, \\cite{chen2014conservative,levine2005membrane,teigen2009diffuse}.\n\nIf the finite element method is a discretization of choice for the bulk problem, then it is natural to consider a finite element method for surface PDE as well. However,  depending on application, desired conservation properties, available software or personal experience, other discretizations such as finite volume or finite difference methods can be preferred for the PDE posed in the volume.\nOne possibility to reuse the same mesh for the surface PDE is to consider  a diffuse-interface approach. Alternatively, instead of smearing the interface, one may extend the PDE off the  surface to a narrow band containing the surface\nin such a way that the restriction of the extended PDE solution back to the (sharp) surface solves the original equation on this surface. Further a conventional discretization is built for the resulting volume PDE in the narrow band~ \\cite{BCOS01,olshanskii2016narrow}. The methods based on such extensions, however, increase the number of the active degrees of freedom for the discrete surface problem, may lead to degenerated PDE, need numerical boundary conditions and require smooth surfaces with no  geometrical singularities.\n\nThe present  paper develops a numerical method based on the sharp-interface representation, which uses a finite volume method to discretize the bulk PDE. Our goal is (i) to allow the surface to overlap with the background mesh in an arbitrary way, (ii) to avoid \\rev{building regular surface triangulation}, (iii)  to avoid any extension of the surface PDE to the bulk domain.  To accomplish these goals, we combine the monotone (i.e. satisfying the discrete maximum principle) finite volume method on  general meshes  \\cite{Lipnikov:12,ChernyshenkoFV7:14}   with the trace finite element method on octree meshes from \\cite{chernyshenko2015adaptive}.\nIn the octree TraceFEM one considers the bulk finite element space of piecewise trilinear globally continuous functions\nand further uses the restrictions (traces) of these functions to the surface. These traces are further used in a variational formulation of the surface PDE. Effectively, this results in the integration of the standard polynomial functions over the (reconstructed) surface.  Only degrees of freedom from the cubic cells cut by the\nsurface are active for the surface problem. Surface parametrization is not required, no surface mesh is built,  no PDE extension off the surface is needed.  We shall see that the resulting hybrid FV--FE method is very robust with respect to the position of surfaces against the background mesh and is well suited for handling non-smooth surfaces and surfaces given implicitly.\n\nOne application of interest is the numerical simulation of the contaminant  transport and diffusion in fractured porous media. In this application, transport and diffusion along fractures are often modeled by PDEs posed on a set of piecewise-smooth surfaces, see, e.g.,~\\cite{alboin2002modeling,fumagalli2013reduced,maryvska2005numerical,therrien1996three}; \\rev{see also \\cite{alboin2002modeling,frac1,frac2,frac3} for a similar dimension reduction approach in simulation of flow in fractured porous media}.\nMonotone (satisfying the DMP) finite volume methods on general meshes is the appealing tool for the solution of\nequations for solute concentration in the porous matrix, see, e.g., ~\\cite{ChernyshenkoFV7:14,Droniou:11,GaoWu:13,KapyrinFV7:14,LePotier:08,Lipnikov:12,ShengYuan:11}\n(further references can be found in \\cite{Droniou:14,FVCA7:14}).\n\n However, a straightforward application of this technique to model transport and diffusion along a fracture would require fitting the mesh or triangulating  the surface. For a large and complex net of fractures cutting through the porous matrix \\rev{this is a difficult task \\cite{DPRF}, and} an efficient method avoids mesh fitting and surface triangulations. \\rev{\n Recently, extended finite element method  approximations have been extensively studied in transport and flow problems in fractured porous media, see the review \\cite{flemisch2016review} and references therein.  In XFEM, one also avoids fitting of the background mesh to a fracture, but a separate mesh  is still required to represent the fracture.\n Besides the use of FV for the matrix problem, the approach in the present paper differs from those found in existing XFEM literature in the way the surface problem is discretized.}\n\nWhile the present technique can be applied for tetrahedral or more general polyhedral  tessellations of the bulk domain, we consider octree grid with cubic cells here. This choice is not \\textit{ad hoc}. Indeed, the Cartesian structure and built-in hierarchy of octree grids makes mesh adaptation, reconstruction and data access fast and easy. For these reasons, octree meshes became a common tool in \nin computational mechanics and  several octree-based solvers  are available in the open source scientific computing software, \\cite{DEAL2,Pop03}. However, an octree grid provides \\rev{at most} the first order (staircase) approximation of a general geometry. Allowing the surface to cut through the octree grid in an arbitrary way overcomes this issue, but challenges us with the problem of building efficient bulk--surface discretizations.   This paper demonstrates that the hybrid TraceFEM -- non-linear FV method complements the advantages of using octree grids by delivering \\rev{more accurate treatment of the surface PDE problem}.\n\nThe remainder of the paper is organized as follows. In section~\\ref{s_setup} we recall the system of differential equations, boundary and interface conditions, which models the coupled bulk--interface (or ``matrix--fracture'' in the context of flows in porous media) advection--diffusion problem. Section~\\ref{s_hybrid} gives the details of the hybrid discretization.\nAfter laying out the main ideas behind the method, we discuss the non-linear monotone FV method for the bulk and the TraceFEM for\nthe surface equations, and further we introduce the required coupling.  Section~\\ref{s_numer} presents the results of several numerical experiments with steady analytical solutions on  smooth and piecewise smooth branching surface. We also show the results of numerical simulation of the propagating front of solute concentration through fractured porous media. \n\n\n\n\n\\section{Mathematical model}\\label{s_setup}\nIn this section we recall the mathematical model of the contaminant  diffusion and transport in fractured  porous media.\nAssume the given bulk  domain $\\Omega \\subset \\mathbb{R}^3$ and a piecewise smooth surface $\\Gamma\\subset\\Omega$. The surface $\\Gamma$ may have several connected components. If  $\\Gamma$ has a boundary, we assume that ${\\partial\\Gamma} \\subset{\\partial\\Omega} $.\nThus, we have the subdivision $\\overline{\\Omega}=\\cup_{i=1,\\dots,N}\\overline{\\Omega}_i$ into simply connected subdomains $\\Omega_i$ such that $\\overline{\\Omega}_i\\cap\\overline{\\Omega}_j\\subset\\Gamma$, $i\\neq j$.\n\nIn each  $\\Omega_i$, we assume a given Darcy velocity field of the fluid $\\mathbf w_i(\\mathbf x)$, $\\mathbf x \\in \\Omega_i$.\nBy $\\mathbf w_\\Gamma(\\mathbf x)$, $\\mathbf x \\in \\Omega_\\Gamma$, we denote the velocity field tangential to $\\Gamma$ having the physical meaning of the flow rate through the cross-section of the fracture.  Consider an agent that is soluble in the fluid and transported by the flow in the bulk and along the fractures. The fractures are modeled by the surface $\\Gamma$.\nThe solute \\emph{volume} concentration (i.e., the one in the bulk domain $\\Omega$) is denoted by $u$, $u_i=u|_{\\Omega_i}$. The solute \\emph{surface} concentration along $\\Gamma$ is denoted by $v$.\nChange of the concentration happens due to convection by the  velocity fields $\\mathbf w_i$ and $\\mathbf w_\\Gamma$, diffusive fluxes in $\\Omega_i$, diffusive flux on $\\Gamma$, as well as the fluid exchange and diffusion flux between the fractures and the porous matrix.\nThese coupled processes can be modeled by the following system of equations~\\cite{alboin2002modeling}, in subdomains,\n\\begin{equation} \\label{diffeq1}\n\\left\\{\n\\begin{aligned}\n \\phi_i\\frac{\\partial u_i}{\\partial t} + \\mbox{\\rm div}(\\mathbf w_i  u_i- D_i \\nabla u_i) &= f_i \\quad \\text{in}~ \\Omega_i,\\\\\n u_i&=v \\quad \\text{on}~ {\\partial\\Omega} _i\\cap\\Gamma,\n \\end{aligned}\n\\right.\n \\end{equation}\nand on the surface,\n\\begin{equation} \\label{diffeq2}\n \\phi_\\Gamma\\frac{\\partial v}{\\partial t} + \\mbox{\\rm div}_\\Gamma (\\mathbf w_\\Gamma v - d D_\\Gamma \\nabla_\\Gamma v) =  F_\\Gamma(u)+ f_\\Gamma  \\quad \\text{on}~~\\Gamma,\n \\end{equation}\nwhere we employ the following notations:  $\\nabla_\\Gamma$,  $\\mbox{\\rm div}_\\Gamma$  denote the surface tangential gradient and divergence\n \\begin{wrapfigure}{r}{0.4\\textwidth}\n\\vspace{-20pt}\n  \\begin{center}\n  \n    \\caption{2D illustration of our notation for a domain with triple fraction.\n    }\n    \\label{fig:Schema}\n  \\end{center}\n  \\vspace{-20pt}\n  \\vspace{1pt}\n\\end{wrapfigure}\noperators; $F_\\Gamma(u)$ stands for the  net flux of the solute per surface area due to fluid leakage and hydrodynamic dispersion;   $f_i$ and $f_\\Gamma$ are given source terms in the subdomains and in the fracture;  $D_i$ denotes the diffusion tensor in the porous matrix; the surface diffusion tensor is $D_\\Gamma$.\nBoth $D_i$, $i=1,\\dots,N$, and $D_\\Gamma$ are symmetric and positive definite; $d>0$ is the fracture width coefficient; $\\phi_i>0$ and $\\phi_\\Gamma>0$ are the constant porosity coefficients for the bulk and the fracture.\n\nThe total surface flux $F_\\Gamma(u)$ represents the contribution of the bulk to the solute transport in the fracture. The mass balance at  $\\Gamma$ leads to the equation\n\\begin{equation}\\label{Flux}\n F_\\Gamma(u)= [-D \\mathbf n \\cdot \\nabla u + (\\mathbf n \\cdot\\mathbf w)u]_\\Gamma,\n\\end{equation}\nwhere $\\mathbf n$ is a unit normal vector at $\\Gamma$,\n$[w(\\mathbf x)]_\\Gamma = \\lim\\limits_{\\eps\\to0} w(\\mathbf x-\\eps\\mathbf n)-\\lim\\limits_{\\eps\\to0} w(\\mathbf x+\\eps\\mathbf n)$, $\\mathbf x\\in\\Gamma$,\ndenotes the jump of $w$ across $\\Gamma$ in the direction of $\\mathbf n$.\n\nIf $\\Gamma$ is  \\textit{piecewise} smooth, then we need further conditions on the edges.\nAssume an edge $\\mathcal{E}$ is shared by  $M$  smooth components $\\Gamma_i\\subset\\Gamma$.\nLet $v_{j}=v$ on $\\Gamma_j$, while $\\mathbf w_{\\Gamma,j}=\\mathbf w_{\\Gamma}$, $d_j=d$, $D_{\\Gamma,j}=D_\\Gamma$ on $\\Gamma_j$, and $\\mathbf n_{{\\partial\\Gamma} ,j}$ is the outward  normal vector to ${\\partial\\Gamma} _j$ in the plane tangential to $\\Gamma_j$, cf. Figure~\\ref{fig:Schema}.\nThe conservation of fluid mass yields\n\\begin{equation}\\label{cont_w}\n\\sum_{j=1}^M\\mathbf w_{\\Gamma,j}\\cdot\\mathbf n_{{\\partial\\Gamma} ,j}=0\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\nWe assume the continuity of concentration over $\\mathcal{E}$,\n\\begin{equation}\\label{cont_v}\nv_{1}=\\dots=v_{M}\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\nWe also assume the conservation of solute flux over the edge. Thanks to \\eqref{cont_w} and \\eqref{cont_v}, this yields the condition:\n\\begin{equation}\\label{cont_F}\n\\sum_{j=1}^M d_j(D_{\\Gamma,j}\\mathbf n_{{\\partial\\Gamma} ,j})\\cdot\\nabla_\\Gamma v_j=0\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\n\nFinally, we prescribe Dirichlet's boundary conditions for the concentration $u$ and $v$ on ${\\partial\\Omega} _D$ and ${\\partial\\Gamma} _D$ and homogeneous Neumann's boundary conditions on ${\\partial\\Omega} _N$ and ${\\partial\\Gamma} _N$, respectively, with $\\overline{{\\partial\\Omega} }=\n\\overline{{\\partial\\Omega} _D}\\cup\\overline{{\\partial\\Omega} _N}$ and $\\overline{{\\partial\\Gamma} }=\\overline{{\\partial\\Gamma} _D}\\cup\\overline{{\\partial\\Gamma} _N}$. Initial conditions are given by the known concentration $u_0$ and $v_0$ at $t=0$. We have\n\\begin{equation} \\label{bc}\n\\left\\{\n\\begin{aligned}\n D_i \\mathbf n_{\\partial\\Omega}  \\cdot \\nabla u&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\\\\n u&=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n u|_{t=0}&=u_0 \\quad \\text{in}~ \\Omega,\\\\\n \\end{aligned}\n\\right.\\qquad \\left\\{\n\\begin{aligned}\nD_{\\Gamma}\\mathbf n_{{\\partial\\Gamma} }\\cdot\\nabla_\\Gamma v&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\\\\n v&=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D,\\\\\n v|_{t=0}&=v_0 \\quad \\text{on}~~\\Gamma.\n \\end{aligned}\n\\right.\n \\end{equation}\n\n\\begin{remark}\\rm Bulk--surface coupled systems of advection-diffusion PDEs appear in different applications,\ne.g.  in multiphase fluid dynamics \\cite{GrossReuskenBook} and biological applications \\cite{bonito2011dynamics}. In these and other models, the continuity of the concentration over the embedded surface (second equation in \\eqref{diffeq1}) may \\rev{be replaced by} another suitable   constitutive equation for modeling of the surface adsorption\/desorption.\nFor fluid--fluid interfaces or biological membranes, one often assumes that the surface passively evolves with the flow, and hence there is no contribution of the advective flux to the total flux $F_\\Gamma(u)$ on $\\Gamma$. A standard model for the diffusive flux between the surface and the bulk, cf.~\\cite{Ravera}, is as follows:\n\\begin{equation} \\label{eq4}\n-D_i \\mathbf n \\cdot \\nabla u_i  =  k_{i,a} g_i(v) u_i - k_{i,d} f_i(v),\\quad\\text{on}~\\Gamma,\n\\end{equation}\nwith $k_{i,a}$, $k_{i,d}$ positive adsorption and desorption coefficients that describe the kinetics.  Basic choices for $g$, $f$ are the following:\n\\[\n  g(v)=1, \\quad f(v)=v \\quad \\text{(Henry)}\\quad \\text{or}\\quad g(v)=1- \\frac{v}{v_{\\infty}},\\quad f(v)=v \\quad \\text{(Langmuir)},\n\\]\nwhere $v_\\infty$ is a constant that quantifies the maximal concentration on $\\Gamma$. Further options are given in \\cite{Ravera}.\nOften in literature on the two-pase flows the Robin condition in \\eqref{eq4} is replaced by the ``instantaneous'' adsorption and desorption condition\n\\begin{equation} \\label{eq4a}\nk_{i,a} g_i(v) u_i = k_{i,d} f_i(v),\\quad\\text{on}~\\Gamma,\n\\end{equation}\nThese interface conditions can be also handled through obvious modifications  of our numerical method. \\rev{We include one numerical example with \\eqref{eq4a} and Henry law in Section~\\ref{s_numer}.} At the same time, treating evolving interfaces needs more developments and is not considered here.\n\\end{remark}\n\n\\section{Hybrid finite volume -- finite element method}\\label{s_hybrid}\n\n\\subsection{Summary of the method}\n\nAssume a Cartesian background mesh with cubic cells. We allow local refinement of the mesh by sequential division\nof any cubic cell into 8 cubic subcells. This leads to a grid with an octree hierarchical  structure. This mesh gives the tessellation   $\\mathcal T_h$ of the computational domain $\\Omega$, $\\overline{\\Omega}=\\cup_{T\\in\\mathcal T_h} \\overline{T}$.\nThe surface  $\\Gamma\\subset\\Omega$ cuts through the mesh in an arbitrary way. For the \\textit{purpose of numerical integration},\ninstead of $\\Gamma$ we consider $\\Gamma_h$, a given polygonal approximation of $\\Gamma$. If $\\Gamma$ has a curvature, then $\\Gamma_h$ is reconstructed as a second order approximation of $\\Gamma$. We shall describe the reconstruction algorithm further in the section. We assume that similar to $\\Gamma$, the reconstructed surface $\\Gamma_h$ divides $\\Omega$ into $N$ subdomains $\\Omega_{i,h}$, and ${\\partial\\Gamma} _h\\subset{\\partial\\Omega} $. We do not \\rev{assume} any restrictions on how $\\Gamma_h$ intersects the background mesh.\n\nThe induced tessellation of  $\\Omega_{i,h}$ can be considered as a subdivision of the volume into general polyhedra.\nHence, for the transport and diffusion in the matrix we apply  a non-linear FV method devised on general polyhedral meshes in \\cite{Lipnikov:12,ChernyshenkoFV7:14}, which is monotone and has compact stencil. The trace of the background mesh on $\\Gamma_h$ induces \\rev{a} `triangulation' of the fracture, which is very irregular, and so we do not use it do build a discretization method.\nTo handle transport and diffusion along the fracture, we first consider finite element space of piecewise trilinear functions\nfor the \\textit{volume} octree mesh $\\mathcal T_h$. We further, formally, consider the restrictions (traces) of these background functions on $\\Gamma_h$ and use them in a finite element integral  form over $\\Gamma_h$.\nThus the irregular triangulation of $\\Gamma_h$ is used \\textit{for numerical integration only}, while the trial and test functions are tailored to the background \\textit{regular} mesh. \\rev{Available analysis and numerical experience suggest} that the \\rev{approximation and convergence} properties of this trace finite element method\n\\rev{depend only on the mesh size and refinement strategy for}  the background mesh, and they are independent on how $\\Gamma_h$ intersects $\\mathcal T_h$. The TraceFEM was devised and first analysed in \\cite{ORG09} and extended for the octree meshes in \\cite{chernyshenko2015adaptive}.  A natural way to couple two approaches is to use the\nrestriction of the background FE solution on $\\Gamma_h$ as the boundary data for the FV method and to compute the FV two-side fluxes on $\\Gamma_h$ to provide the source terms for the surface discrete equation. We provide details of each of these steps in sections~\\ref{s_FV}--\\ref{s3} below.\n\n\n\\subsection{Reconstructed surface}\\label{s_rec}\nThe reconstructed surface $\\Gamma_h$ is a $C^{0,1}$ surface that can be partitioned in planar triangular segments:\n\\begin{equation} \\label{defgammah}\n \\Gamma_h=\\bigcup\\limits_{K\\in\\mathcal{F}_h} K,\n\\end{equation}\nwhere $\\mathcal{F}_h$ is the set of all  triangular segments $K$.\nWithout loss of generality we assume that for any $K\\in\\mathcal{F}_h$ there is only \\textit{one}\ncell $T_K\\in\\mathcal{T}_h$ such that $K\\subset T_K$ (if $K$ lies on a face shared by two cells, any of these\ntwo cells can be chosen as $T_K$).\n\nIn practice, we construct $\\Gamma_h$ as follows. For each connected piece of $\\Gamma$ let $\\phi$ be a Lipschitz-continuous level set function, such that $\\phi(\\mathbf x)=0$ on $\\Gamma$. We set $\\phi_h=I(\\phi)$  a nodal interpolant of $\\phi$ by a  piecewise trilinear continuous function with respect\nto the octree grid $\\mathcal{T}_h$.  Consider the zero level set of $\\phi_h$,\n\\[\n\\widetilde{\\Gamma}_h:=\\{\\mathbf x\\in\\Omega\\,:\\, \\phi_h(\\mathbf x)=0 \\}.\n\\]\nIf $\\Gamma$  is smooth, then  $\\widetilde{\\Gamma}_h$ is an approximation to $\\Gamma$ in the following sence:\n\\begin{equation}\\label{eq_dist}\n\\mbox{dist}(\\Gamma,\\rev{\\widetilde{\\Gamma}_h})\\le ch^2_{\\rm loc},\\qquad |\\mathbf n(\\mathbf x)-\\mathbf n_h(\\tilde\\mathbf x)|\\le c h_{\\rm loc},\n\\end{equation}\nwhere $\\mathbf x$ is the closest point on $\\Gamma$ for $\\tilde\\mathbf x\\in\\widetilde{\\Gamma}_h$ and $h_{\\rm loc}$ is the local mesh size.\nWe note that in some applications, $\\phi_h$ is computed from a solution of a discrete indicator function equation (e.g., in the level set or the volume of fluid methods), without any direct knowledge of $\\Gamma$.\n\n\\begin{figure}\n  \\begin{center}a)\n    \\includegraphics[width=0.31\\textwidth]{.\/mesh3d_rotsin.jpg}b)\n    \\includegraphics[width=0.31\\textwidth]{.\/mesh2d_rotsin.jpg} c)\n    \\includegraphics[width=0.29\\textwidth]{.\/mesh2d_rotsin_zoom.jpg}\n    \\caption{a) Example of a  bulk domain with a fracture. In this example, the background mesh is refined near the fracture; b) The reconstructed $\\Gamma_h$; c) The zoom-in of the induced surface triangulation $\\mathcal F_h$.}\n    \\label{fig1}\n  \\end{center}\n\n\\end{figure}\n\\rev{Note that $\\widetilde{\\Gamma}_h$ is still not completely suitable for our purposes, since $\\phi_h$ is \\emph{trilinear}\nand so numerical integration over its zero level is not straightforward. Therefore, we next build a suitable polygonal approximation of $\\widetilde{\\Gamma}_h$ which is our final $\\Gamma_h$.}\nOnce $\\phi_h$ is computed, we recover $\\Gamma_h$ by the cubical marching squares method from \\cite{MCM2} (a variant of the very well-known marching cubes method). The method provides a triangulation of $\\widetilde{\\Gamma}_h$ within each\ncube such that \\rev{$\\Gamma_h$ is  continuous over cubes interfaces}, the number of triangles within each cube is finite and bounded\nby a constant independent of $\\widetilde{\\Gamma}_h$ and a number of refinement levels. Moreover, the vertices of triangles from $\\mathcal{F}_h$ are lying on $\\widetilde{\\Gamma}_h$. This final discrete surface $\\Gamma_h$ is still an approximation of $\\Gamma$ in the sense of \\eqref{eq_dist}. A example of bulk domain with embedded surface and background mesh is illustrated in Figure~\\ref{fig1}.\n\nNote that the resulting triangulation $\\mathcal{F}_h$  is \\emph{not} necessarily  regular, i.e. elements from $T$ may have very small internal angles and the size of neighboring triangles can vary strongly. Thus, $\\Gamma_h$ is not \\rev{ a regular triangulation of $\\Gamma$.} The surface triangulation $\\mathcal{F}_h$ is used only to define quadratures in the finite element method, while approximation properties of the method depend on the volumetric octree mesh.\n\n\n\n\\subsection{Monotone finite volume method} \\label{s_FV}\nFirst we consider a FV method for the advection-diffusion equation \\eqref{diffeq1} in each subdomain $\\Omega_{i,h}$.\nLet $\\mathcal T_{i,h}$ be the tessellation  of $\\Omega_{i,h}$ into non-intersected polyhedra, which is induced by overlapping $\\Omega_{i,h}$ and the background mesh $\\mathcal T_h$.   Since the background mesh is the octree Cartesian, each element $T\\in\\mathcal T_{i,h}$ is either a cube, if it lies in the interior of $\\Omega_{i,h}$, or a cut cube, if ${\\partial\\Omega} _{i,h}$ intersects a background cell from $\\mathcal T_h$. We assume the octree grid is gradely refined, i.e. the sizes of two neighbouring elements of $\\mathcal T_h$ can differ at most by a factor of two.  Such octree grids are also known as balanced. The method applies for unbalanced octrees, but in our experiments we use balanced grids.\nFor the balanced grid,  each interior cell may have from 6 to 24 neighboring cells (cells sharing a face). In the FV method we treat such cells as a polyhedra with up to 24 faces.   Since we reconstruct $\\Gamma$ inside each cell as a triangulated surface without holes, the cut cell from $\\mathcal T_{i,h}$ can be treated as a general polyhedral element as well. By $\\mathcal F_{i,h}$ we denote the set of all faces of polyhedra from $\\mathcal T_{i,h}$.\n\nThe FV discretization below is applied to each subdomain $\\Omega_i$ separately, so we will skip in this section the  redundant index $i$ for the concentration, coefficients and the flow vector field in $\\Omega_i$. Note that ${\\partial\\Omega} _{i,D}$\nincludes the fracture part of the boundary of ${\\partial\\Omega} _i$.\n\nAs the first step,  we assume a time discretization (say, the implicit Euler method) and consider the mixed form of  \\eqref{diffeq1}  and  boundary conditions\n\\begin{equation}\\label{lap}\n\\begin{array}{rclll}\n  \\mathbf q = \\mathbf w u - D \\nabla u, \\quad\n  \\tilde\\phi\\,u+\\mbox{\\rm div } \\mathbf q  &=& f \\quad     & \\mbox{~in} & \\Omega_i,  \\\\\n       u &=& \\tilde{u}_D                     & \\mbox{~on} & {\\partial\\Omega} _{i,D} ,  \\\\\n       -D \\mathbf n_{\\partial\\Omega}  \\cdot \\nabla u &=&0 & \\text{~on}& {\\partial\\Omega} _{i,N},\n\\end{array}\n\\end{equation}\nwhere the right hand side $f$ accounts for the source term and for the values of concentration from the previous time step, $\\tilde\\phi$ is the scaled \\rev{(by the reciprocal of time step)} porosity coefficient\\rev{, and ${\\partial\\Omega} _{i,D}$ includes ${\\partial\\Omega} _i\\cap\\Gamma$, where $\\tilde{u}_D=v$}.\n\nFor a cell $T\\in\\mathcal T_{i,h}$, $\\mathbf x_{T}$ denotes the barycenter of $T$, and $u_T$ denotes the averaged concentration. We formally assign $u_T$ to $\\mathbf x_T$.\nIntegrating the mass balance equation \\eqref{lap} over $T$ and using the divergence theorem, we obtain:\n\\begin{equation}\\label{green2}\n \\tilde\\phi|T|u_T +\\sum\\limits_{\\mathcal F \\in \\partial T} \\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F  =  \\int_T f \\,{\\rm d} x,\n  \\qquad\n  \\mathbf q_{\\mathcal F} = \\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  \\mathbf q \\,{\\rm d} s,\n\\end{equation}\nwhere $\\mathbf q_{\\mathcal F} \\cdot \\mathbf n_\\mathcal F$ is the averaged normal flux across face $\\mathcal F$,  and $\\mathbf n_\\mathcal F$  is the normal vector\non $\\mathcal F$ pointing outward for $T$; $|\\mathcal F|$ ($|T|$) denotes the area (volume) of $\\mathcal F$ ($T$).\nThe Dirichlet boundary data on faces $\\mathcal F\\in \\partial\\Omega_D$ will be accounted in the scheme via boundary faces concentration values\n$u_\\mathcal F = \\frac{1}{|\\mathcal F|}\\int_{\\mathcal F}  u_D \\,{\\rm d} s$. We assume that $u_\\mathcal F$ are assigned to barycenters of faces.\n Enforcing homogeneous Neumann boundary conditions on faces from ${\\partial\\Omega} _{i,N}$ is straightforward, for $\\mathcal F\\in  \\partial T\\cap{\\partial\\Omega} _{N}$ the normal flux $\\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F$ in \\eqref{green2} is  set to 0.\n\nIn the conventional cell-centered FV method, the normal flux $\\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F$ is replaced by its discrete counterpart $\\mathbf q_{\\mathcal F,h} \\cdot \\mathbf n_\\mathcal F$, which is computed\nfrom cell concentrations $u_T$ and boundary data $u_\\mathcal F$. For simplicity of presentation we shall omit subscript $h$ in notations of the discrete flux.\nThe discrete flux is the combination of the diffusive and convective fluxes and we discretize them separately\nfollowing  \\cite{Lipnikov:10,Lipnikov:12,NikitinVassilevski:10}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=.45\\textwidth]{.\/quadtreecut_MPFA3.pdf}\n\\caption{For the 2D case, the figure illustrates our constructions for the approximation of the directional derivative $\\mathbf l_{\\mathcal F}\\cdot\\nabla u$ on the face (edge in 2D) $\\mathcal F=\\overline{T_+}\\cap\\overline{T_-}$. Bold dots\nshow the barycenters $\\mathbf x_T$ of the cells from $\\mathcal T_{1,h}$ and $\\mathcal T_{2,h}$. The mean values of the concentration are assigned to these barycenters.}\n\\label{fig:dupl}\n\\end{figure}\n\nFor $T\\in\\mathcal T_{i,h}$, we define $\\omega(T):=\\{T'\\in\\mathcal T_{i,h}\\,|\\,\\mbox{area}(\\overline{T}'\\cap \\overline{T})\\neq 0\\}$, the set of all neighboring cells of $T$, and\n $\\omega_{\\partial}(T):=\\{F\\in\\mathcal F_{i,h}\\,|\\,F\\rev{\\subset} \\partial T\\cap \\partial\\Omega_{i,D}\\}$, the set all faces of $T$ with prescribed Dirichlet data.  For $T\\in\\mathcal T_{i,h}$, the set of points $\\mathcal{P}$ collects all barycenters of the elements from $\\omega(T)$ and $\\omega_{\\partial}(T)$.  Furthermore, for each $T\\in\\mathcal T_{i,h}$ we define the bundle of vectors, $\\mathbf v(T):=\\{\\mathbf t\\in\\mathbb{R}^3\\,|\\,\\mathbf t=\\mathbf y-\\mathbf x_T,~\\mathbf y\\in \\mathcal{P}(T)\\}$.\n\nConsider an arbitrary internal face $\\mathcal F$ shared by two cells $T_+, T_-$ from $\\mathcal T_{i,h}$ and assume that $\\mathbf n_\\mathcal F$ points from $T_+$ to $T_-$. We introduce the co-normal vector $\\mathbf l_{\\mathcal F}= D \\mathbf n_\\mathcal F$.\nVector $\\mathbf l_{\\mathcal F}$  can make a nonzero angle with $\\mathbf n_\\mathcal F$ in the case of an anisotropic diffusion tensor.\nTo define the discrete {\\em diffusive} flux on $\\mathcal F$, we first\nconsider three vectors $\\mathbf t_{i}^+\\in \\mathbf v(T_+)$, $i=1,2,3$,\nsuch that for the co-normal vector $\\mathbf l_{\\mathcal F} = D \\mathbf n_f$ we have\n\\begin{equation}\\label{ab}\n  \\mathbf l_{\\mathcal F} =\n  \\alpha_+~ \\mathbf t_{1}^+ + \\beta_+~ \\mathbf t_{2}^+ + \\gamma_+~ \\mathbf t_{3}^+,\n\\end{equation}\nwith non-negative coefficients $\\alpha_+$, $\\beta_+$ and $\\gamma_+$. Such a triplet can be always found,  (in some rare pathological situations, one has to expand $\\mathcal{P}(T_+)$ slightly, cf. \\cite{DanilovVassilevski:09}).\n\nThe normal flux is the directional derivative along the co-normal vector $\\mathbf l_+ := \\mathbf l_{\\mathcal F}$, and hence\nit can also be represented as the linear combination of three derivatives along $\\mathbf t_{i}^+$. The latter  are approximated by central differences (may reduce to one side differences near Dirichlet boundaries). Thus, we get\n$\\mathbf q_\\mathcal F\\cdot\\mathbf n_F\\approx q_+$,\n\\begin{equation}\\label{ab2}\n q_+=\n  \\alpha'_{+}~ (u_+ - u_{+,1}) + \\beta'_{+}~ (u_+ - u_{+,2}) +\n  \\gamma'_{+}~ (u_+ - u_{+,3}),\\qquad u_+=u(\\mathbf x_{T_+}),~ u_{+,i}=u(\\mathbf x_{T_+}+\\mathbf t_{i}^+),\n\\end{equation}\nwhere coefficients $\\alpha'_{+}, \\beta'_{+}, \\gamma'_{+}$ are computed from $\\alpha$, $\\beta$, $\\gamma$ in  \\eqref{ab}\nfor the cell $T_+$, using the simple  scaling with $|\\mathbf t_{i}^+| \/ |\\mathbf l_{\\mathcal F}|$.\nFor the same co-normal vector one has  another decomposition based on $\\mathbf v(T_-)$ vector bundle,\n$\n\\mathbf l_- := -\\mathbf l_{\\mathcal F} =\n  \\alpha_-~ \\mathbf t_{1}^- + \\beta_-~ \\mathbf t_{2}^- + \\gamma_-~ \\mathbf t_{3}^-$, $\\mathbf t_{i}^-\\in \\mathbf v(T_-)$. This decomposition yields another approximation,  $\\mathbf q_\\mathcal F\\cdot\\mathbf n_F\\approx q_-$:\n\\begin{equation}\\label{ab3}\n  q_- =\n  \\alpha'_{-}~ (u_- - u_{-,1}) + \\beta'_{-}~ (u_- - u_{-,2}) +\n  \\gamma'_{-}~ (u_- - u_{-,3}),\\qquad u_-=u(\\mathbf x_{T_-}),~ u_{-,i}=u(\\mathbf x_{T_-}+\\mathbf t_{i}^-),\n\\end{equation}\nwith non-negative coefficients $\\alpha'_-$, $\\beta'_-$ and $\\gamma'_-$. Figure~\\ref{fig:dupl} illustrates the construction in 2D.\n\nNow we can take a linear combination of \\eqref{ab2} and \\eqref{ab3} with non-negative coefficients $\\mu_+$ and $\\mu_-$:\n\\begin{equation}\\label{two-flux}\n  \\mathbf q_{\\mathcal F} \\cdot \\mathbf n_{\\mathcal F} = \\mu_+ q_+ + \\mu_- (- q_-).\n\\end{equation}\nThe discrete flux \\eqref{two-flux}  approximates the differential one if   $\\mu_+$, $\\mu_-$  satisfy\n\\begin{equation}\n\\label{mupm1}\n\\mu_+ + \\mu_- = 1.\n\\end{equation}\nFollowing \\cite{Lipnikov:12}, to construct the {\\it monotone}  FV discretization, we set  both representations of the flux equal:\n\\begin{equation}\n\\label{fvdmp_mus}\n  \\mu_+ q_+ = - \\mu_- q_-.\n\\end{equation}\nIf $| q_+| = |q_-| = 0$, then the solution of \\eqref{mupm1},\\eqref{fvdmp_mus} in not unique. In this case we choose $\\mu_+ = \\mu_- = 1\/2$.\nOtherwise, the solution is given by\n\\begin{equation*}\n\\mu_+ = \\frac{q_-}{q_- - q_+}, \\qquad \\mu_- = \\frac{q_+}{q_+ - q_-}.\n\\end{equation*}\nIf $q_+ q_- > 0$, we avoid   potentially degenerate case by applying the modification from~\\cite{ShengYuan:11} \\rev{(see also formulas in \\cite{Lipnikov:12}, p.~374)}.\nWe note that the resulting multi-point flux approximation is nonlinear and compact, i.e. the stencil includes the values of concentration only from neighboring cells.\n\nTo define the normal component of the discrete \\textit{advective} flux\n$\\mathbf q_{\\mathcal F,a} =\\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  u\\mathbf w \\,{\\rm d}s$,\nwe adopt the nonlinear  upwind approximation (subscript $h$ is again omitted for the sake of notation):\n\\begin{equation}\\label{vRvR}\n  \\mathbf q_{\\mathcal F,a} \\cdot \\mathbf n_{\\mathcal F} = w_{\\mathcal F}^+  \\mathcal R_{T^+}(\\mathbf x_{\\mathcal F}) + w_{\\mathcal F}^-  \\mathcal R_{T^-}(\\mathbf x_{\\mathcal F}),\n\\end{equation}\nwhere\n$$\nw_{\\mathcal F}^+ =\\frac12 (w_{\\mathcal F}+ |w_{\\mathcal F}|), \\quad w_{\\mathcal F}^- =\\frac12 (w_{\\mathcal F}- |w_{\\mathcal F}|), \\quad w_{\\mathcal F} = \\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  \\mathbf w\\cdot\\mathbf n_{\\mathcal F} \\,{\\rm d} s,\n$$\n$\\mathcal R_T$ is a linear reconstruction of the concentration\nover cell $T$ which depends on the concentration values from neighboring cells.\n\n\nOn each cell $T$, the linear reconstruction is defined by\n\\begin{equation}\\label{reconst}\n\\mathcal R_T (\\mathbf x) = \\left\\{ \\begin{array}{ll} u_T + \\mathcal L_T \\mathbf g_T \\cdot (\\mathbf x-\\mathbf x_{T}), & \\mathbf x\\in T, \\\\\n                                       0, & \\mathbf x\\notin T, \\end{array} \\right.\n\\end{equation}\nwhere $\\mathbf g_T$ denotes the gradient of the  linear reconstruction of concentration in $\\mathbf x_T$, and $\\mathcal L_T$ is a slope limiting operator. The gradient is recovered from the best affine least-square fit for $u_h$ over a subset of barycenter nodes and, possibly, the boundary data nodes from cells neighboring $T$.\nThe slope limiting operator $\\mathcal L_T$ is introduced to avoid non-physical extrema. It provides the smallest possible changes of the reconstructed least-square slope. Details can be found in  \\cite{Lipnikov:10,Lipnikov:12,NikitinVassilevski:10}.\n\n\nReplacing fluxes in equations \\eqref{green2} by their numerical approximations, we\nobtain a system of nonlinear equations\n\\begin{equation} \\label{sol:system}\n  \\Phi\\mathbf U+ \\mathbf M(\\mathbf U)\\, \\mathbf U = \\mathbf F(\\mathbf U),\n  \\qquad\n  \\mathbf M(\\mathbf U) = \\mathbf M_{dif}(\\mathbf U) + \\mathbf M_{adv}(\\mathbf U),\n\\end{equation}\nwith a diagonal matrix $\\Phi$. For any fixed vector $\\mathbf V$, $\\mathbf M(\\mathbf V)$ is a square sparse matrix,  $\\mathbf F(\\mathbf V)$ is a right-hand side vector.\nMatrix $\\mathbf M_{dif}$ is an M-matrix which has  diagonal dominance in rows.\nThe stencil of this matrix is compact, each row contains non-zero off-diagonal entries corresponding mainly (and in most cases only) to degrees of freedom at the cells sharing a face with the current cell.\n For a  cubic uniform mesh and the Poisson equation, the matrix $\\mathbf M_{dif}$ corresponds to the conventional seven-point stencil.\nAlthough matrix $\\mathbf M_{adv}$ has no diagonal dominance in rows,\nit can be shown, cf.~\\cite{Lipnikov:12}, that the solution to \\eqref{sol:system}\nsatisfies the discrete maximum principle.\n\n\n\\subsection{The trace finite element method} \\label{s_FEM}\n\nConsider now the volumetric finite element space of all piecewise trilinear continuous functions with respect to the bulk octree mesh $\\mathcal{T}_h$:\n\\begin{equation}\n V_h:=\\{v_h\\in C(\\Omega)\\ |\\ v|_{S}\\in Q_1~~ \\forall\\ S \\in\\mathcal{T}_h\\},\\quad\\text{with}~~\n Q_1=\\mbox{span}\\{1,x_1,x_2,x_3,x_1x_2,x_1x_3,x_2x_3,x_1x_2x_3\\}.\n \\label{e:2.6}\n\\end{equation}\nThe surface finite element space is \\textit{the space of traces on $\\Gamma_h$ of all piecewise trilinear continuous functions with respect to the outer triangulation $\\mathcal{T}_h$}  defined as follows\n\\begin{equation}\n V_h^{\\Gamma}:=\\{\\psi_h\\in H^1(\\Gamma_h)\\ |\\ \\exists ~ v_h\\in V_h\\  \\text{such that }\\ \\psi_h=v_h|_{\\Gamma_h}\\}.\n\\label{e:fem-space}\n\\end{equation}\n\nGiven the surface finite element space $V_h^{\\Gamma}$,  the finite element\ndiscretization of \\eqref{diffeq2} is as follows:   Find $v_h\\in V_h^{\\Gamma}$ such that $v_h|_{\\partial\\Gamma_{D,h}}=v_D^h$ and\n\\begin{multline}\n\\int_{\\Gamma_h}\\left(  \\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}w_h +  d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\, +(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h) w_h\\right)\\, + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, \\rev{w_h}v_h \\mathrm{d}\\mathbf{s}_h\\\\ =\\int_{\\Gamma_h}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h}) w_h\\, \\mathrm{d}\\mathbf{s}_h \\label{FEM}\n\\end{multline}\nfor all $w_h\\in V_h^{\\Gamma}$, s.t. $w_h|_{\\partial\\Gamma_{D,h}}=0$ . Here $\\mathbf{w}_h$, $v_D^h$, $d_h$, $D_{\\Gamma,h}$ and $f_{\\Gamma,h}$ are  the problem data lifted from $\\Gamma$ to $\\Gamma_h$, in the case if $\\Gamma\\neq\\Gamma_h$. The bulk domain contributes through the flux $F_{\\Gamma,h}(u_h)$, which is reconstructed from the numerical concentration in the porous matrix.\n\nSimilar to the plain Galerkin finite element for  advection-diffusion equations the method \\eqref{FEM} is prone to instability unless mesh is sufficiently fine such that the mesh Peclet number is less than one.\nFollowing \\cite{ORXimanum}, we consider the SUPG stabilized TraceFEM.  The stabilized formulation reads:\n Find $v_h\\in V_h^{\\Gamma}$ such that\n\\begin{multline}\n \\int_{\\Gamma_h}\\left(  \\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}w_h +  d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\, +(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h) w_h\\right)\\, + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, \\rev{w_h}v_h \\mathrm{d}\\mathbf{s}_h \\\\\n  +\\sum_{\\rev{T}\\in\\mathcal{F}_h}\\delta_K\\int_{K}(\\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}-\\rev{d\\mbox{\\rm div}_{\\Gamma_h}D_{\\Gamma,h}\\nabla_{\\Gamma_h}}v_h + \\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, v_h)\\,\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} w_h\\, \\mathrm{d}\\mathbf{s}_h\\\\ =\\int_{\\Gamma_h}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h}) w_h\\, \\mathrm{d}\\mathbf{s}_h + \\sum_{K\\in\\mathcal{F}_h}\\delta_K\\int_{K}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h})(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} w_h)\\, \\mathrm{d}\\mathbf{s}_h\\quad \\forall~ w_h\\in V_h^{\\Gamma}. \\label{FEM_SUPG}\n\\end{multline}\n\\rev{For the definition of $K\\in\\mathcal{F}_h$, $T_K\\in\\mathcal T_h$ we refer to section~\\ref{s_rec}.}\nThe stabilization parameter  $\\delta_K$ depends on $K \\subset T_K$. The side length of the cubic cell $T_K$ is denoted by $h_{T_K}$.  Let $\\displaystyle \\mathsf{Pe}_K:=\\frac{h_{T_K} \\|\\mathbf{w}_h\\|_{L^\\infty(K)}}{2\\eps}$\nbe the cell Peclet number.\nWe take\n\\begin{equation}\n {\\delta_K}=\n\\left\\{\n\\begin{aligned}\n&\\frac{\\delta_0 h_{T_K}}{\\|\\mathbf{w}_h\\|_{L^\\infty(K)}} &&\\quad \\hbox{ if } \\mathsf{Pe}_K> 1,\\\\\n&\\frac{\\delta_1 h^2_{T_K}}{\\eps}  &&\\quad \\hbox{ if } \\mathsf{Pe}_K\\leq 1,\n\\end{aligned}\n\\right. \\label{e:2.10}\n\\end{equation}\nwith some given positive constants  $\\delta_0,\\delta_1\\geq 0$.\n\n\nFor the matrix--vector representation of the TraceFEM one uses the nodal basis of the bulk finite element space $V_h$ rather than tries to construct a basis in $V_h^{\\Gamma}$.  This convenient choice, however, has some consequences. In general, the restrictions to $\\Gamma_h$ of the outer nodal basis functions on $\\Gamma_h$ can be linear dependent or (in most cases) almost linear dependent.\nThis and small cuts of background cells lead to badly conditioned mass and stiffness matrices. In recent\nyears stabilizations have been developed which are easy to implement and result in\nmatrices with acceptable condition numbers, see the overview in \\cite{TraceFEM}.\nIn this paper we use the ``full gradient'' stabilization of the TraceFEM~\\cite{Alg2,Reusken2014}. In this variant of the method, one modifies the surface diffusion part of the  method~\\eqref{FEM} to include the normal part of the gradient:\n\\[\n\\int_{\\Gamma_h}d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\,\\mathrm{d}\\mathbf{s}_h\\quad\\text{yields to}\\quad\n\\int_{\\Gamma_h}d_h D_{\\Gamma,h}\\nabla v_h\\cdot\\nabla w_h\\,\\mathrm{d}\\mathbf{s}_h.\n\\]\nWe note that the method remains consistent on smooth surfaces (up to second order geometric errors), since the true surface solution extended off the surface along normal directions satisfies both variational formulations on $\\Gamma$.   The modification improves algebraic properties of the (diagonally scaled) stiffness matrix of the method~\\cite{Reusken2014}. The full-gradient method  uses the background finite element space $V_h$ instead of the surface finite element space $V_h^\\Gamma$ in \\eqref{FEM}. However, practical implementation of both methods uses the frame of all bulk finite element nodal basis functions $\\phi_i\\in V_h$ such that  $\\mbox{supp}(\\phi_i)\\cap\\Gamma_h\\neq\\emptyset$. Hence the active degrees of freedom  in both methods are the same. The stiffness matrices are, however, different.\n\n\n\\subsection{Coupling between discrete bulk and surface equations}\\label{s3}\n\nThe equations in the bulk and on the surface are coupled through the boundary condition $u_i=v$ on $\\partial\\Omega_{i,h}\\cap\\Gamma_h$ (second equation in \\eqref{diffeq1}) and the net flux $F_{\\Gamma_h}(u)$ on $\\Gamma_h$, which stands as the source term in the surface equation \\eqref{diffeq2}. On $\\Gamma_h$ the solution $v_h$ is defined as a trace of the background finite element piecewise trilinear function. The averaged value\n of $v_h$ is computed on each surface triangle $K\\in\\mathcal F_h$ using a standard quadrature rule. These values assigned to the barycenters of $K$ from $\\mathcal F_h$ serve as the Dirichlet boundary data for the FV method on $\\Gamma_h$.\n The discrete diffusive and convective fluxes are assigned to barycenters of all faces on $\\mathcal T_{i,h}$, $i=1,\\dots,N$.\nSince each triangle $K\\in\\mathcal F_h$ is a face for two cells $T_i\\in\\mathcal T_{i,h}$ and  $T_j\\in\\mathcal T_{j,h}$, $i\\neq j$,\nthe diffusive and convective fluxes are assigned to $K$ from both sides of $\\Gamma_h$.\nThe discrete net flux $F_{\\Gamma_h}(u_h)$ at the barycenter of $K$ is computed as the jump of the fluxes over $K$.\nIn the TraceFEM this value is assigned to all $\\mathbf x\\in K$, and numerical integration is  done over all surface elements $K\\in\\mathcal F_h$ to compute the right-hand side of the algebraic system.\n\nTo satisfy all (discretized) equations and boundary conditions we iterate between the bulk FV and surface FE solvers on each time step.\nWe assume an implicit time stepping method (in experiments we use backward Euler). This results in the following system on each time step.\n\\begin{equation} \\label{system}\n\\left\\{\n\\begin{aligned}\n \\mathcal L u:= \\tilde\\phi u + \\mbox{\\rm div}(\\mathbf w  u- D \\nabla) u &= \\hat{f} \\quad \\text{in}~ \\Omega\\setminus\\Gamma,\\\\\n u&=v \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla u=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad u&=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\mathcal L_\\Gamma v:=\\tilde\\phi_\\Gamma v + \\mbox{\\rm div}_\\Gamma (\\mathbf w_\\Gamma v - d D_\\Gamma \\nabla_\\Gamma v) &=  F_\\Gamma(u)+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla v=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad v&=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D,\n \\end{aligned}\n\\right.\n\\end{equation}\nthe right hand sides $\\hat{f}$ and $\\hat{f}_\\Gamma$ account for the  solution values at the previous time step. Note that condition \\eqref{cont_v} is satisfied by the construction of trace spaces in the finite element method\nand condition \\eqref{cont_F} is accounted weakly by the TraceFEM variational formulation.\n\n\\rev{\nWe solve the coupled system \\eqref{system} by the fixed point method: Given ${u}^0,{v}^0$, the initial guess,  we iterate for $k=0,1,2,\\dots$ until convergence: \\\\\nStep 1: Solve for ${u}^{k+1}$,\n\\begin{equation} \\label{step1}\n\\left\\{\n\\begin{aligned}\n \\mathcal L {u}^{k+1}&= \\hat{f}~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad {u}^{k+1}={v}^{k} \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla {u}^{k+1}&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad {u}^{k+1}=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\end{aligned}\n\\right.\n\\end{equation}\nStep 2: Solve for ${v}^{\\rm aux}$ and update for ${v}^{k+1}$ with a relaxation parameter $\\omega$,\n\\begin{equation} \\label{step2}\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma {v}^{\\rm aux}&=  F_\\Gamma(u^{k+1})+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla_\\Gamma {v}^{\\rm aux}&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad {v}^{\\rm aux}=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D\\\\\n {v}^{k+1}&=\\omega{v}^{\\rm aux}+(1-\\omega) {v}^{k},\\quad \\omega\\in(0,1],\n \\end{aligned}\n\\right.\n\\end{equation}\n}\n\n\\begin{remark}\\rm \\label{rem1} \\rev{Below we show that the fixed point method is equivalent to a preconditioned Richardson iteration for the discrete Poincar\\'{e}--Steklov operator. Assume that $\\mathcal L$ is linear (this is true for our differential model, but\nthe particular FV discretization applied here is actually non-linear).\nLet's} split $u=u_0+\\hat{u}$, $v=v_0+\\hat{v}$, where $u_0,\\,v_0$ satisfy\n\\[\n\\left\\{\n\\begin{aligned}\n\\mathcal L u_0&= \\hat{f}~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad u_0=0 \\quad \\text{on}~ \\Gamma,\\\\\n D\\mathbf n_{\\partial\\Omega} \\cdot\\nabla u_0&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad u_0=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\n  \\end{aligned}\n\\right.\\qquad\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma {v}_0&=  0  \\quad \\text{on}~~\\Gamma,\\\\\n D_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla {v}_0&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad {v}_0=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D\n \\end{aligned}\n\\right.\n\\]\nNow the iterations \\eqref{step1}--\\eqref{step2} can be written in terms of  $\\hat{u}$ and $\\hat{v}$ parts of the bulk and surface  concentrations:\n\\begin{equation} \\label{step1a}\n\\begin{split}\n\\left\\{\n\\begin{aligned}\n \\mathcal L \\hat{u}^{k+1}&= 0~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad \\hat{u}^{k+1}=\\hat{v}^{k} \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla \\hat{u}^{k+1}&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad \\hat{u}^{k+1}=0 \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\end{aligned}\n\\right. \\\\\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma \\hat{v}^{\\rm aux}&=  F_\\Gamma(u^{k+1})+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla_\\Gamma \\hat{v}^{\\rm aux}&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad \\hat{v}^{\\rm aux}=0 \\quad \\text{on}~ {\\partial\\Gamma} _D\\\\\n \\hat{v}^{k+1}&=\\omega\\hat{v}^{\\rm aux}+(1-\\omega) \\hat{v}^{k},\\quad \\omega\\in(0,1].\n \\end{aligned}\n\\right.\n\\end{split}\n\\end{equation}\nNow we note that $\\hat{u}$ is a (generalized) harmonic extension of $\\hat{v}$  on $\\Omega\\setminus\\Gamma$ and $S_\\Gamma\\,:\\,\\hat{v}\\to F_\\Gamma(\\hat{u})$ is the  Dirichlet to Neumann (discrete) Poincar\\'{e}--Steklov operator. Using this notation,\none can write the surface equation for $\\hat{v}$ in the compact operator form,\n\\begin{equation} \\label{PS}\n(\\mathcal L_{\\Gamma,0}-S_\\Gamma) \\hat{v}= \\hat{F}\\quad \\text{on}~~\\Gamma,\\quad\\text{with}~~ \\hat{F}:=F_\\Gamma(u_0)+ \\hat{f}_\\Gamma.\n\\end{equation}\nWe use zero index in $\\mathcal L_{\\Gamma,0}$ to stress that the operator accounts for homogenous boundary conditions on ${\\partial\\Gamma} $. It is easy to see that \\eqref{step1a} is the Richardson iterative process for the surface\nequation \\eqref{PS}, with the preconditioner $W=\\mathcal L_{\\Gamma,0}^{-1}$ and the relaxation parameter $\\omega$:\n\\begin{equation} \\label{Richards}\n\\hat{v}^{k+1}=\\hat{v}^{k}-\\omega\\,W\\left(\\, (\\mathcal L_\\Gamma-S_\\Gamma) \\hat{v}^k-\\hat{F}\\right),\\quad k=0,1,2,\\dots.\n\\end{equation}\nFrom \\eqref{Richards} we see that a more efficient iterative process based on a different choice\nof preconditioner and employing  Krylov subspaces may be feasible \\rev{(if $\\mathcal L$ is non-linear one may consider Anderson's mixing to accelerate convergence)}.  However, we do not pursue this topic further in this paper.\n\\end{remark}\n\n\n\\section{Numerical results and discussion}\\label{s_numer}\nThis section collects several numerical examples, which demonstrate the accuracy and capability of the hybrid method.\n We perform a series of tests, where we simulate steady and time-dependent solutions  in a bulk domain with an imbedded  fracture. \\rev{We also include an a example with a smooth curved surface (a sphere) embedded in a bulk domain and a given analytical solution for a surface-bulk problem with Henry interface condition.}\n \\rev{To measure the error we shall use $L^2$, $H^1$ and $L^\\infty$ surface and volume norms.\nFor the computed solutions $u_h,v_h$ and true solutions $u,v$, these norms are defined below. In the volume, we set\n\\[\n\\begin{split}\n&\\mbox{err}_{L^\\infty(\\Omega)}:=\\max_{T\\in\\mathcal T_h}|u_h(\\mathbf x_T)-u(\\mathbf x_T)|,\\quad \\mbox{err}_{L^2(\\Omega)}:=\\left(\\sum_{T\\in\\mathcal T_h}\\text{vol}(T)|u_h(\\mathbf x_T)-u(\\mathbf x_T)|^2\\right)^{\\frac12},\\\\ &\\mbox{err}_{H^1(\\Omega)}:=\\left(\\sum_{T\\in\\mathcal T_h}\\text{vol}(T)|\\nabla I(u_h)(\\mathbf x_T)-\\nabla u(\\mathbf x_T)|^2\\right)^{\\frac12},\n\\end{split}\n\\]\nwhere $I(u_h)$ is the $P_1$ least-square interpolant to the values of $u_h$ in barycenters of the cells from $\\omega(\\mathbf x_T)\\cap\\Omega_i$, for $T\\in\\Omega_i$. Over the surface, we set\n\\[\n\\mbox{err}_{L^\\infty(\\Gamma)}:=\\max_{\\Gamma}|v_h-v^e|,\\quad \\mbox{err}_{L^2(\\Gamma)}:=\\|v_h-v^e\\|_{L^2(\\Gamma)},\\quad\n\\mbox{err}_{H^1(\\Gamma)}:=\\|\\nabla_{\\Gamma_h}v_h-\\nabla_{\\Gamma_h}v\\|_{L^2(\\Gamma)},\n\\]\nwhere $v^e$ is the  extension of $v$ from $\\Gamma$ to $\\Gamma_h$ along normal directions to $\\Gamma$.\n}\n\n\n\n\n\\subsection{Steady analytical solution for a  triple fracture problem}\\label{sec_ex2}\n\nOur next experiment deals with the coupled surface--bulk diffusion problem in the domain $\\Omega = [0,1]^3$ with\nan embedded piecewise planar $\\Gamma$. We design $\\Gamma$ to model a branching fracture.  In the basic model,\n$\\Gamma=\\Gamma(0)$ consists of three planar pieces,\n\\[\\Gamma(0) = \\Gamma_{12}\\cup\\Gamma_{13}\\cup\\Gamma_{23},\\quad \\Gamma_{ij}=\\overline{\\Omega_i}\\cap\\overline{\\Omega_j}~~i\\neq j,\n\\]\nsuch that\n\\[\n\\Omega_{1}=\\{\\mathbf x\\in\\Omega\\,|\\, x<\\frac12~\\text{and}~ y>x\\},\\quad \\Omega_{2}=\\{\\mathbf x\\in\\Omega\\,|\\, x>\\frac12~\\text{and}~y>x-1\\},\\quad\n\\Omega_{3}=\\Omega\\setminus(\\overline{\\Omega}_{1}\\cup\\overline{\\Omega}_2).\n\\]\nThis subdivision is illustrated in Figure~\\ref{fig:triple} (left). The pieces $\\Gamma_{ij}$ belong to certain planes of symmetry\nfor the cube, and so the induced triangulation of $\\Gamma(0)$ and the cut cells in the bulk domain are all quite regular.\nTo model a generic situation when $\\Gamma$ cuts through the background mesh in an arbitrary way, we  consider other\ntessellations of $\\Omega = [0,1]^3$ into three subdomains by a surface $\\Gamma(\\alpha)$. The surface $\\Gamma(\\alpha)$ is obtained from $\\Gamma(0)$ by applying the clockwise rotation by the angle $\\alpha$ around the axis $x=z=0.5$. We take $\\alpha=20^o$ and\n$\\alpha=40^o$, the resulting tessellations of $\\Omega$ are illustrated in  Figure~\\ref{fig:triple} (middle and right pictures).\nMore precisely, we define\n\\[\n\\Gamma(\\alpha)=\\{\\mathbf x\\in\\Omega\\,|\\, \\mathbf y\\in\\Gamma(0),~\\mathbf y-\\mathbf x_0=\\mathcal{Q}_\\alpha(\\mathbf x-\\mathbf x_0)\\},~~\\text{with}~\n\\mathcal{Q}_\\alpha=\\begin{bmatrix}\\cos\\alpha&0&-\\sin\\alpha\\\\ 0&1&0\\\\ \\sin\\alpha&0&\\cos\\alpha\\end{bmatrix},~ \\mathbf x_0=({\\footnotesize\\frac12},0,{\\footnotesize\\frac12})^T.\n\\]\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.31\\textwidth]{.\/mesh3d_triple_grey3.jpg} \\includegraphics[width=0.36\\textwidth]{.\/explodeTrip_grey.jpg} \\includegraphics[width=0.31\\textwidth]{.\/meshRot40_explode_grey.jpg}\n\\caption{\\label{fig:triple}\nThe figure illustrates the bulk domain with uniform mesh and the fracture. On the left picture the fracture is set orthogonal to the $xy$-plane, while on the middle and right pictures the fracture is rotated by 20 and 40 degrees.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.32\\textwidth]{.\/mesh2d_grey.jpg}  \\includegraphics[width=0.32\\textwidth]{.\/mesh2d_rot20_2.jpg}\n\\includegraphics[width=0.32\\textwidth]{.\/mesh2d_rot40_grey2.jpg}\n\\caption{\\label{fig:frac}\nThe figure illustrates the induced surface mesh on the fracture, when it cuts through the uniform bulk mesh in different ways.}\n\\end{center}\n\\end{figure}\n\n\nSimilar to the series of numerical experiments with the embedded spherical $\\Gamma$, here we set   the source terms $f_i$ and $f_\\Gamma$ and the boundary conditions such that the solution to the stationary problem \\eqref{diffeq1}--\\eqref{bc} is known.  To define the  solution $\\{v,u\\}$ solving the stationary equations \\eqref{diffeq1}--\\eqref{bc}, we first introduce\n\\begin{equation}\\nonumber\n\\psi_1 = \\left\\{\n\\begin{array}{lr}\n 16(y-\\frac12)^4,& y>\\frac12\\\\\n 0, & y \\le \\frac12\\\\\n\\end{array}\n\\right.,\\quad\\psi_2 = x -y,\\quad \\psi_3 = x + y - 1.\n\\end{equation}\nWe define the solution of the basic model problem ($\\alpha=0$)\n\\begin{equation}\\nonumber\nu(\\mathbf x)=\\left\\{\n \\begin{array}{rl r}\n  & \\sin(2\\pi z)\\cdot\\psi_2(\\mathbf x) \\cdot\\phi_3(\\mathbf x) & \\mathbf x\\in \\Omega_1,\\\\\n  & \\sin(2\\pi z)\\cdot\\psi_1(\\mathbf x) & \\mathbf x\\in \\Omega_2,\\\\\n  & \\sin(2\\pi z)2x\\cdot\\psi_1(\\mathbf x) & \\mathbf x\\in \\Omega_3,\n \\end{array}\n \\right.\\qquad v=u|_{\\Gamma(0)}.\n\\end{equation}\nNote that the constructed exact solution is continuous across $\\Gamma(0)$, but the normal derivatives are discontinuous.\nOther parameters in \\eqref{diffeq1}--\\eqref{diffeq2} are set to be  $\\mathbf{w}=\\mathbf{w}_\\Gamma=0$, $\\phi_1=\\phi_2=\\phi_\\Gamma=0$, $D_1=D_2=D_\\Gamma=I$, and $d=1$. For the problem setup with the rotated fracture, $\\alpha>0$ we set the exact solution\n$v_\\alpha(\\mathbf x)=v(\\mathbf y)$, $u_\\alpha(\\mathbf x)=u(\\mathbf y) $, with $\\mathbf y=\\mathcal{Q}_\\alpha(\\mathbf x-({\\small\\frac12},0,{\\small\\frac12})^T)$.\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 0$.\n\\label{tab:triple0}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&855&\t6.374e-3     &        &4.214e-1&          & 3.920e-2&      \\\\\n&7410&\t 1.698e-3    & 1.84   &1.631e-1&1.36      & 1.276e-2& 1.56 \\\\\n&61620&\t4.235e-4     & 1.97   &6.193e-2&1.39      & 3.506e-3& 1.83 \\\\\n&502440& 1.044e-4    & 2.00   &2.348e-2&1.40      & 1.129e-3& 1.62 \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&232& 8.469e-3       &         &2.914e-1 &          &9.280e-3&     \\\\\n&1242&2.003e-3\t       & 1.79  &1.387e-1 & 0.92     &2.779e-3& 1.44\\\\\n&5662&5.588e-4\t       & 1.84  &6.874e-2 & 1.01     &1.217e-3& 1.09\\\\\n&24102&1.791e-4       &  1.64  &3.395e-2 & 1.02     &5.181e-4& 1.18\\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 20$.\n\\label{tab:triple20}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&965&6.319e-3&               &4.208e-1&        &3.754e-2  &    \\\\\n&7872&1.805e-3      & 1.79   &1.661e-1& 1.34   &1.280e-2  &1.55\\\\\n&63592& 5.623e-4    & 1.80   &6.371e-2& 1.38   & 3.411e-3 &1.90\\\\\n&510390&1.602e-4    & 1.81   &2.442e-2& 1.39   & 1.146e-3 &1.57\\\\\\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&321& 7.792e-3    &       &2.694e-1&        &2.716e-2&     \\\\\n&1692&2.084e-3    & 1.59  &1.240e-1& 1.12   &5.400e-3& 1.94\\\\\n&7944&7.019e-4    & 1.41  &6.291e-2& 0.98   &2.001e-3& 1.29\\\\\n&{\\color{black}33272}&{\\color{black}2.441e-4}   & {\\color{black}1.52}  &{\\color{black}3.173e-2}& {\\color{black}0.99}   &{\\color{black}7.217e-4}& {\\color{black}1.47}\\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 40$.\n\\label{tab:triple40}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&991&\t 5.934e-3   &       & 4.080e-1&      &  3.783e-2 &     \\\\\n&7996&\t 1.700e-3   & 1.80  & 1.621e-1& 1.33 &  1.276e-2 & 1.56\\\\\n&64046&  4.907e-4   & 1.80  & 6.263e-2& 1.37 &  3.515e-3 & 1.86\\\\\n&512258& 1.503e-4   & 1.82  & 2.541e-2& 1.39 &  1.237e-3 & 1.61\\\\  \\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&353&     8.167e-3 &       &2.709e-1&     & 2.696e-2 &      \\\\\n&1932&\t  2.146e-3 & 1.66  &1.275e-1& 1.09& 5.566e-3 & 1.85 \\\\\n&8766&    7.115e-4 & 1.59  &6.279e-2& 1.02& 2.063e-3 & 1.31 \\\\\n&{\\color{black}36676}&   {\\color{black}2.538e-4} & {\\color{black}1.49}  &{\\color{black}3.121e-2}& {\\color{black}1.01}& {\\color{black}7.251e-4} & {\\color{black}1.51} \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\label{s_ex2}\n\nThe numerical results for this coupled problem with the triple fracture problem are reported in Tables~\\ref{tab:triple0}--\\ref{tab:triple40}.  We observe stable convergent results for $\\alpha=0$ as well as for more general case of $\\alpha>0$.   An interesting feature of this problem is that the surface $\\Gamma$ is\nonly piecewise smooth.  The bulk grid is not fitted to the internal edge $\\mathcal{E}=\\Gamma_{12}\\cap\\Gamma_{13}\\cap\\Gamma_{23}$,\nand hence the tangential derivatives of $v$ are discontinuous inside certain cubic cells from $\\mathcal T_h^\\Gamma$.\n\\rev{Therefore, a kink in $v$ cannot be represented by the finite element approximation.} This \\rev{may} result in  \\rev{a}  reduction of convergence order. \\rev{Both the performance of the FV method for cut  cells (cut cells inherit a regular structure from the background mesh for $\\alpha=0$, but are very irregular for $\\alpha>0$ ) and the presence of the kink influences the observed convergence rates. }\n\n\\begin{table}[h!]   \\rev{\n\\begin{center}\n\\caption{Iteration numbers in \\eqref{step1}--\\eqref{step2} for the steady problem example in section~\\ref{sec_ex2}.\n\\label{t_conv}}\\smallskip\n\\small\n\\begin{tabular}{r|lll}\\hline\nref. level. & $\\alpha=0$ & $\\alpha=20^o$ & $\\alpha=40^o$ \\\\ \\hline\\\\[-2ex]\n0&\t22   & 74  &24    \\\\\n1&\t29   & 90  &32    \\\\\n2&\t212  & 325 &228    \\\\\n3& 782   & 917 &851   \\\\ \\hline\n\\end{tabular}\n\\end{center}    }\n\\end{table}\n\n\\rev{\nFinally, Table~\\ref{t_conv} shows the performance of the fixed-point iteration~\\eqref{step1}--\\eqref{step2}. We set $\\omega=1$ and take $u^0=0$, $v^0=0$. The stopping criterion was the relative\ndecrease of the Euclidian norm of  both surface and bulk equations residuals by a factor of $10^4$ (a stronger convergence criterion was not found to improve solution accuracy). On each outer iteration, the surface linear subproblem was solved by exact factorization, while a few Picard iterations with exact factorization of linearized problem were done to solve the bulk system in \\eqref{step1}.   The solver does not scale in an optimal way with respect to the mesh size and more research is needed to improve its performance, cf. Remark~\\ref{rem1}. We postpone this topic for the future research. We also note that for time dependent problems studied  below including time-dependent terms and taking initial guess to be the solution from the previous time step improves convergence of \\eqref{step1}--\\eqref{step2} a lot,\nand we typically need 1 or 2 iterations on each time step.}\n\n\n\\subsection{Propagating front in the porous medium with triple fracture}\n\\definecolor{linkcolor}{HTML}{799B03}\n\\definecolor{urlcolor}{HTML}{799B03}\n\n\\hypersetup{pdfstartview=FitH, linkcolor=linkcolor,urlcolor=urlcolor, colorlinks=true}\n\n\\begin{figure}[h!]\n\\begin{center}\na)~~\\includegraphics[width=0.4\\textwidth]{.\/diff3d_0.png}\\qquad b)~~\\includegraphics[width=0.4\\textwidth]{.\/diff3d.png}\\\\\nc)~~\\href{www.math.uh.edu\/~molshan\/OLSH\/pdf_refs\/pdf_ref_video_outx4.html}{\\includegraphics[width=0.4\\textwidth]{.\/conv3d_0.png}}\\qquad d)~~\\includegraphics[width=0.4\\textwidth]{.\/conv3d.png}\\\\\n\\caption{\\label{fig:prop}\nThe figures illustrates the propagating front of the concentration in the fracture and in the bulk: Pictures a), b) show dominating diffusion case, while  c), d) show dominating convection case. Pictures a) and c) \\rev{show} snapshots \\rev{of} the computed solution at time t=0.018, while pictures b) and d) snapshots the computed  solution at time t=\\rev{0.033}. Click on picture~c) to run full animation of the experiment.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\na)~~\\href{www.math.uh.edu\/~molshan\/OLSH\/pdf_refs\/pdf_ref_video_out_new_higher.html}{\\includegraphics[width=0.4\\textwidth]{.\/diff3drot.png}}\\qquad b)~~\\includegraphics[width=0.4\\textwidth]{.\/conv3drot.png}\n\\caption{\\label{fig:prop1}\nThe figures illustrates the propagating front of the concentration in the fracture and in the bulk, with $\\alpha=20^o$: Picture a) shows dominating diffusion case, picture b) shows dominating convection case;  both at time t=0.033. Click on picture~a) to run full animation of the experiment.}\n\\end{center}\n\\end{figure}\n\nIn the last series of experiments we compute the time dependent solution of \\eqref{diffeq1}--\\eqref{bc}. The bulk domain $\\Omega$\nand the fracture $\\Gamma$ are the same as in the previous experiment in section~\\ref{s_ex2}.\nAt  time\n$t_0=0$ we set $u(t_0)=0$ in $\\Omega$ and  $v(t_0)=0$ on $\\Gamma$. On the face $\\{y=1\\}$ of the cube  we prescribe the constant concentration of a contaminant, while on other parts on the boundary the  diffusion flux is  set equal zero. Thus in \\eqref{bc}, we have\n\\[\n\\begin{split}\n\\partial\\Omega_D&=\\partial\\Omega\\cap\\{y=1\\},\\quad\\partial\\Omega_N=\\partial\\Omega\\setminus\\partial\\Omega_D,\\quad \\partial\\Gamma_D=\\partial\\Gamma\\cap\\{y=1\\},\\quad \\partial\\Gamma_N=\\partial\\Gamma\\setminus\\partial\\Omega_D,\\\\  u_D&=1,\\quad v_D=1,\\quad u_0=0,\\quad \\text{and}\\quad v_0=0.\n\\end{split}\n\\]\nThe time independent velocity field transports the contaminant in the bulk and along the fractures.\nWe set\n\\[\n\\begin{split}\n\\mathbf w_i&=2\\kappa(0,-1,0)^T,~~i=1,2,3,\\quad\\text{in}~\\Omega\\\\\n\\mathbf w_\\Gamma&=5\\kappa(0,-1,0)^T~~\\quad\\text{in}~\\Gamma_{12},\\quad\n\\mathbf w_\\Gamma=\\kappa\\mathcal{Q}_\\alpha({\\footnotesize\\frac{1}{\\sqrt{2}}},-{\\footnotesize\\frac{1}{\\sqrt{2}}},0)^T\n~~\\quad\\text{in}~\\Gamma_{23},\\quad\n\\mathbf w_\\Gamma=\\kappa\\mathcal{Q}_\\alpha(-{\\footnotesize\\frac{1}{\\sqrt{2}}},-{\\footnotesize\\frac{1}{\\sqrt{2}}},0)^T~~\\quad\\text{in}~\\Gamma_{13},\n\\end{split}\n\\]\nwhere $\\kappa\\ge0$ is a parameter. One easily verifies the condition \\eqref{cont_w} on the edge $\\mathcal{E}=\\Gamma_{12}\\cap\\Gamma_{13}\\cap\\Gamma_{23}$. Other parameters are set to be\n\\[\nD_1=D_2=0.1\\,I,\\quad d=1,\\quad D_\\Gamma=I, \\quad \\phi_1=\\phi_2=\\phi_\\Gamma=1.\n\\]\n\n\n\n\nThe computed solutions for $\\kappa=1\/8$ (diffusion dominated case) and $\\kappa=8$ (convection plays a significant role)  are illustrated in Figures~\\ref{fig:prop}--\\ref{fig:prop1}. The fracture angle parameter $\\alpha$ was set to $0$ and $20$ degrees, \\rev{respectively}. \n\n\nFor this problem, the exact solution is not known. The computed solution occurs to be physically reasonable. We see no sign of spurious oscillations. \\rev{As expected, the contaminant propagates faster along the fractures.}\n\n\\subsection{Contaminant transport along the fracture.}\n\nIn this test we place  a continuous contaminant source on the upstream boundary of the fracture.\n\\begin{wrapfigure}{r}{0.45\\textwidth}\n\\vspace{-20pt}\n  \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{.\/mesh16rot.jpg\n    \\caption{The matrix and fracture in the test with contaminant transport along the fracture.}\n    \\label{fig:Front2}\n  \\end{center}\n  \\vspace{-20pt}\n  \\vspace{1pt}\n\\end{wrapfigure}\nThe matrix--fracture configuration for this test is shown in Figure~\\ref{fig:Front2}, $\\Omega=(0,1)^3$, and $\\Gamma=\\{\\mathbf x\\in\\Omega\\,:\\,z+\\frac12x=0.51\\}$. The boundary $x=0$\nis inflow, in the fracture the wind is constant $\\mathbf w_\\Gamma=(w_1,0,w_3)$, $|\\mathbf w_\\Gamma|=1$, and the contaminant source occupies  the part of $\\partial\\Gamma$, \\rev{${\\partial\\Gamma} _D=\\{(0,y,0.51)\\,:\\,y\\in(\\frac14,\\frac34)\\}$, $v_D=1$ on ${\\partial\\Gamma} _D$.}\nWe assume that the  porous matrix  is almost impermeable and so we set $\\mathbf w_i=0$ in $\\Omega_i$ (no flow in the rock) and $D_i=10^{-6}I$, $i=1,2$, \\rev{${\\partial\\Omega} ={\\partial\\Omega} _N$}. In the fracture we assume isotropic diffusion with $D_\\Gamma=10^{-4}I$. Other parameters are the same as in the previous test, \\rev{and $v=0$, $u=0$ at $t=0$}. Therefore, we expect that the contaminant transport happens along the fracture with very small diffusion to the porous matrix. This is a bulk--surface variant of a standard test case of numerical solvers for convection--diffusion problems~\\cite{sun2013mathematical},  and one is typically interested in the ability of a method to capture the right position and the shape of the sharp propagating front and avoid spurious oscillations. For a comparison purpose, one may consider the exact solution for the problems posed\nin a half-plane (or half-space) from~\\cite{leij1990analytical,leij1991analytical}. This solution $C(x,y,0)$ is given in~\\eqref{kapanalsol1}, it solves $C_t-D\\Delta C+{C}_{x}=0$ in  $\\widetilde \\Omega = \\{\\rev{(x,y)}\\in\\mathbb{R}^2\\,:\\, x>0\\}$, with the\nboundary condition\n$\nC(0,y,t)=\\left\\{\n\\begin{array}{c}\nc_0, \\textrm{  when } |y|<a,\\\\\n0,  \\textrm{  when } |y|>a,\n\\end{array}\n\\right.\n\nand  initial conditions: $C(x,y,0)= 0$ in $\\widetilde\\Omega$.\n\\begin{equation}\n C(x,y,t)=\\frac{x c_0}{(16 \\pi D)^{\\frac 12}} \\int \\limits_0^t  \\tau^{-\\frac 32}\n\\left\\{\n\\mathbf{erf} \\left[ \\frac{a+y}{(4 D \\tau)^\\frac 12} \\right]\n+\\mathbf{erf} \\left[ \\frac{a-y}{(4 D \\tau)^\\frac 12} \\right]\n\\right\\} \\cdot \\mathbf{exp} \\left[ - \\left( \\frac{x-\\tau}{(4D \\tau)^{\\frac 12}} \\right) ^2 \\right] d\\tau.%\n\\label{kapanalsol1}\n\\end{equation}\nwhere\n\\begin{equation*\n\\mathbf{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\int \\limits_0^x e^{-t^2} dt, \\quad \\mathbf{erfc}(x ) = 1-\\mathbf{erf}(x)= \\frac{2}{\\sqrt{\\pi}} \\int \\limits_x^\\infty e^{-t^2} dt.\n\\end{equation*}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/esol_017_full.jpg}  \\\\\n\\includegraphics[width=0.8\\textwidth]{.\/esol_34_full.jpg}  \\\\\n\\includegraphics[width=0.8\\textwidth]{.\/32sol_full.jpg}\n\\caption{\\label{fig:rot20_3d}\nReference 2D solution  (left) and the fracture component of the computed  solution (right) for the contaminant transport along the fracture test\ncase. The solutions is shown for the times $t=0.17,\\,0.34,\\,0.5$.}\n\\end{center}\n\\end{figure}\n\nWe run our simulations with the uniform background mesh,  $h=\\frac1{32}$, $\\Delta t=10^{-2}$. The fracture cuts through the background mesh as illustrated in  Figure~\\ref{fig:Front2} (for better visualization, this figure shows the background mesh for $h=\\frac1{16}$). The computed solution and the `reference' solution is shown in  Figure~\\ref{fig:rot20_3d} at several time instances. We recall that the coupled problem was solved and the contaminant also diffuses into the bulk, but this bulk diffusion was minor. We observe that the\ncomputed solution well approximates the reference one; the computed front has the correct position and \\rev{is not  smeared too much. Moreover,} we do not observe  overshoots or undershoots in $v_h$.\n\n{\\color{black}\n\\subsection{An example with a spherical drop immersed in a bulk}\nWe include one more test case but now with a different interface condition. This is\nthe instantaneous absorbtion--desorption condition \\eqref{eq4a} with the Henry law to define $g_i$ and $f_i$.\nThis condition is common in the literature to model dissolvable surfactant transport in two-phase flows.\nIn this test from~\\cite{gross2015trace} we consider a prototypical configuration for such models consisting of a spherical drop embedded in a cubic domain. We take $\\Gamma$ to be the unit sphere centered at the origin and $\\Omega = [-1.2,1.2]^3$.\nBy $\\Omega_1$ we denote the interior of $\\Gamma$, so $\\Omega_1$ is the unit ball,  \\rev{$\\Omega_2=\\Omega\\setminus\\overline{\\Omega}_1$}. For the velocity field we take a rotating field in the $x$-$z$ plane: $\\mathbf w=\\frac{1}{10}(z,0,-x)$. This $\\mathbf w$ satisfies   $\\mbox{\\rm div}\\mathbf w=0$ in $\\Omega$ and $\\mathbf w \\cdot \\mathbf n=0$ on $\\Gamma$, i.e. the velocity field is everywhere tangential to the boundary and hence the steady interface is consistent with the kinematic condition: $\\mathbf w \\cdot \\mathbf n$ is equal to the normal velocity of $\\Gamma$ for immersible two-pase fluids, e.g.~\\cite{GrossReuskenBook}. We set $\\mathbf w_i=\\mathbf w|_{\\Omega_i}$ and $\\mathbf w_\\Gamma=\\mathbf w|_{\\Omega_\\Gamma}$.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{.\/new_sphere_solmesh.jpg}\\qquad  \\includegraphics[width=0.4\\textwidth]{.\/new_sphere_3Dsolmeshcut.jpg}\n\\caption{\\label{fig:sphere_loc}\nLeft: Induced surface mesh and the surface component of computed solution.\nRight: Cut of the bulk mesh and the volume component of computed solution.}\n\\end{center}\n\\end{figure}\n\nThe material parameters are chosen as $D_1=0.5$, $D_2=1$, $D_\\Gamma=1$ and $k_{1,a}=0.5$, $k_{2,a}=2$, $k_{1,d}=2$, $k_{2,d}=1$, $d=1$.\nThe source terms $f_i\\in L^2(\\Omega)$, $i=1,2$, and $f_\\Gamma\\in L^2(\\Gamma)$   and  data on ${\\partial\\Omega} $ are taken such  that the exact solution of the  \\textit{stationary} equations \\eqref{diffeq1}--\\eqref{diffeq2} is given by\n\\begin{equation} \\label{exsol}\n \n  v(x,y,z) = 3x^2y - y^3, \\quad\n  u_1(x,y,z) = 2 u_2(x,y,z), \\quad\n  u_2(x,y,z) = e^{1-x^2-y^2-z^2} v(x,y,z).\n\\end{equation}\nSince we solve for a steady-state solution, so we  set $\\phi_1=\\phi_2=\\phi_\\Gamma=0$.\n We prescribe Dirichlet boundary conditions on $\\partial\\Omega$, i.e. $\\partial\\Omega_N=\\emptyset$,   $\\partial\\Gamma_N=\\emptyset$, and $\\partial\\Gamma_D=\\emptyset$ in \\eqref{bc}. Conditions \\eqref{cont_w}--\\eqref{cont_F} for this test case are not relevant, since the surface is globally smooth and has no boundary.\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Convergence of numerical solutions in the experiment with a spherical $\\Gamma$ embedded in a cube.\n\\label{tab:sphere_uni}  }\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&736&\t 3.223e-02 &        & 9.706e-01  &        &  1.072e-01  &      \\\\\n&4920&\t 6.687e-03 &    2.27& 1.555e-01  &    2.64&  3.799e-02  &1.50\\\\\n&36088&  2.005e-03 &    1.74& 5.363e-02  &    1.55&  9.180e-01  &-4.59 \\\\\n&275544& 5.055e-04 &    1.99& 1.825e-02  &    1.56&  2.777e-03  &8.37 \\\\  \\hline\\\\[-2ex]\n\n\\rev{$\\Gamma$}\n&460&      1.670e-02   &    &  2.065e-01&     & 3.863e-02&     \\\\\n&1660&     4.037e-03   &2.05&  9.647e-02& 1.10& 1.060e-02&1.87 \\\\\n&6628&     9.211e-04   &2.13&  4.745e-02& 1.02& 3.881e-03&1.45  \\\\\n&26740&    2.457e-04   &1.91&  2.396e-02& 0.99& 8.875e-04&2.13  \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn this set of experiments we take the sequence of uniform cubic meshes in $\\Omega$, starting with $h=0.3$. The surface $\\Gamma_h$ is reconstructed as described in section~\\ref{s_rec} for $\\phi(\\mathbf x)=1-|\\mathbf x|^2$. The computed solution as well as volume and induced surface meshes are illustrated in Figure~\\ref{fig:sphere_loc}.\nThe computed errors for the bulk and surface concentrations are shown in Table~\\ref{tab:sphere_uni}. For this example, the method\ndemonstrates  optimal convergence: $O(h)$ in the $H^1$ and $O(h^2)$ in the $L^2$ and surface norms. This is consistent with what is known about the convergence of the TraceFEM for linear bulk elements, see e.g. analysis and convergence rates for the same\nexperiment in \\cite{gross2015trace}, where the TraceFEM has been used to discretized equations both on the surface and in the bulk.  For the volume component of the solution, the convergence is close to the second order in the $L^2$ norm and $1.5$ order in the $H^1$ norm.  It is also consistent with the results in \\cite{Lipnikov:12}, where  super-convergence of the method in the $H^1$ norm was observed.  Convergence in the $L^\\infty$-norm is somewhat less regular. We note that the $L^\\infty$ convergence of the TraceFEM and of the non-linear FV method that we used has not been studied before.\n\\medskip\n\nThe aim of the next (final) test is to illustrate the performance of the method for the case of locally refined grids.  The setup is similar to the test with the sphere above, but the coefficients and the known solution are taken different to represent the situation of convection dominated problem with an internal layer. More precisely,  for the velocity field we take  $\\mathbf w=(-y\\sqrt{1-z^2},x\\sqrt{1-z^2},0)$, and set $\\mathbf w_i=\\mathbf w|_{\\Omega_i}$ and $\\mathbf w_\\Gamma=\\mathbf w|_{\\Omega_\\Gamma}$.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{.\/new_sphere_solmesh_cd.jpg}\\qquad  \\includegraphics[width=0.45\\textwidth]{.\/new_sphere_3D_cd.jpg}\n\\caption{\\label{fig:sphere_adap}\nLeft: Induced surface mesh and the surface component of computed solution.\nRight: The bulk mesh and the volume component of computed solution.}\n\\end{center}\n\\end{figure}\n\nThe material parameters are chosen as $D_1=D_2=D_\\Gamma=\\eps$ and $k_{1,a}=0.5$, $k_{2,a}=2$, $k_{1,d}=2$, $k_{2,d}=1$, $d=1$.\nThe source terms $f_i\\in L^2(\\Omega)$, $i=1,2$, and $f_\\Gamma\\in L^2(\\Gamma)$   and  data on ${\\partial\\Omega} $ are taken such  that the exact solution of the  \\textit{stationary} equations \\eqref{diffeq1}--\\eqref{diffeq2} is given by\n\\begin{equation} \\label{exsol}\n \n  v(x,y,z) = {x z}\\mathrm{arctan}\\left(\\frac{2z}{\\sqrt{\\varepsilon}}\\right), \\quad\n  u_1(x,y,z) = 2 u_2(x,y,z), \\quad\n  u_2(x,y,z) = e^{1-x^2-y^2-z^2} v(x,y,z).\n\\end{equation}\nWe take $\\eps=1$ (very smooth solution) and $\\eps=0.01$ (solution has an internal layer along the midplane $z=0$).\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Convergence of numerical solutions in the experiment with a spherical $\\Gamma$ and locally refined mesh.\n\\label{tab:sphere_adap}  }\\smallskip\n\\small\n\\begin{tabular}{r|llllll|llllll}\\hline\n&\\multicolumn{6}{|c|}{$\\varepsilon=1$}&\\multicolumn{6}{|c}{$\\varepsilon=1e-2$}\\\\\n\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate& $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}&&&&&&&&&&&&\\\\\n5840  &   3.993e-03&       &  8.322e-02&        &  1.706e-02&      &   5.666e-02&       &  3.987e-01&        &  1.945e-01&     \\\\\n43552 &   6.706e-04&2.57   &  2.828e-02&1.56    &  4.926e-03& 1.79 &   2.609e-02&1.12   &  2.440e-01&0.71    &  8.403e-02&1.21 \\\\\n318696&   2.200e-04&1.61   &  1.038e-02&1.45    &  2.158e-02&-2.13 &   1.353e-02&0.95   &  1.708e-01&0.52    &  5.003e-02&0.75 \\\\  \\hline\\\\[-2ex]\n\\rev{$\\Gamma$} &&&&&&&&&&&&\\\\\n1500 &       1.916e-03&    &   3.609e-02&     & 6.850e-03&    &    8.353e-03&    &  4.026e-01&     & 4.538e-02&     \\\\\n6740 &       5.106e-04&1.91&   1.710e-02&1.08 & 1.919e-03&1.84&    1.854e-03&2.17&  1.619e-01&1.31 & 1.335e-02&1.76 \\\\\n25988&       1.400e-04&1.87&   8.924e-03&0.94 & 5.624e-04&1.77&    3.848e-04&2.26&  6.532e-02&1.31 & 3.694e-03&1.85 \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nWe build a sequence of locally refined meshes as illustrated in~Figure~\\ref{fig:sphere_adap}.\nFor $\\eps=0.01$ the meshes fit the layer and intend  to capture the sharp variation of the solution.\nWe computed numerical solutions on a sequence of 3 meshes, the second mesh is illustrated in ~Figure~\\ref{fig:sphere_adap}. Each mesh has two levels of refinement in the region $|z|<\\frac18$.\nThe convergence of the method is reported in Table~\\ref{tab:sphere_adap}. The optimal order of convergence is attended for the surface component of the solution, but the FV method in the bulk domain shows lower order convergence for the convection dominated case. We conclude that more studies are required to improve the performance of the FV method on such type of meshes.\n\n\\section{Conclusions} \\label{s_concl} The paper proposed a hybrid finite volume -- finite element method for the coupled bulk--surface systems of PDEs. The distinct feature of the method is that the same background mesh is used to solve equations in the bulk and on the surfaces, and that there is no need to fit this mesh to the embedded surfaces. This makes the approach particularly attractive to treat problems with complicated embedded structures of lower dimension like those occurring in the simulations of flow and transport in fractured porous media. We consider the particular monotone non-linear FV method with compact stencil, but we believe that the approach can be carried over and used with other FV methods on polyhedral meshes (e.g. some of those reviewed in \\cite{Droniou:14}) with possibly better performance in terms of convergence rates. In this paper we treated only diffusion and transport of a contaminant assuming that Darcy velocity is given. Extending the method to computing flows in fractured porous media is in our future plans together with the design of better algebraic solvers, doing research on adaptivity, and adding to the method a fracture prorogation model.\n\n}\n\n\\subsection*{Acknowledgments}\nThe work of the first author (the numerical implementation and the numerical experiments) has been supported  by the Russian Scientific Foundation Grant 17-71-10173, the work of the second author has been supported by the NSF grant 1717516, the work of the third author has been supported by the RFBR grant 17-01-00886.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec1}\n\nWave propagation in heterogeneous complex media has been a subject of intensive research interest in recent years. \nAmong various systems, a large part of the conducted studies has been concentrated in families of one-dimensional ($1$D) continuous and discrete models~\\cite{dyson,disorder_14,izrailev,disorder_15}\nby focusing mainly on the localization properties of both the normal modes of finite systems i.e.,~Anderson localization (AL)~\\cite{anderson}, as well as wave propagation in infinite media.\nThe successful extension of AL to many other systems after it was initially formulated for electronic systems, has opened many research frontiers and applications \\cite{bruinsma,schwartz,lahini,billy,roati}. Experimental results on AL (see e.g.,~Refs.~\\cite{schwartz,lahini,billy,roati,jk}) have stimulated further interest in AL for both quantum and classical systems. \n \nRegarding linear disordered $1$D lattices, among different systems, special attention has been given to the tight binding electron model \\cite{harrison},\nthe linear Klein-Gordon (KG) lattice~\\cite{rogers} and the harmonic lattice \\cite{dyson,Ishii,Ishii_2,Kundu}. These models are not only relevant to various physical systems but also represent the linear limits of seminal nonlinear lattices such as the discrete nonlinear Schr\\\"{o}dinger equation (DNLS), the quartic KG, and the Fermi-Pasta-Ulam-Tsingou (FPUT) lattices~\\cite{fpu,kevre,izrailev,ggg}. Within the context of phononic and photonic lattices, these fundamental models have been adopted to describe a variety of physical systems and more recently, they have been used as a testbed for novel wave phenomena~\\cite{deymer2,flachbook}.\n\nA common route to study the wave properties of disordered lattices is through monitoring the time evolution of initially compact wave-packets. For tight binding and linear KG models, the dynamics after the excitation of such an initial condition is characterized by a transient initial phase of spreading, followed by a phase of total confinement to the system's localization length. The width of the wave-packet is of the order of the maximum localization length~\\cite{loc_len}. On the other hand, for the harmonic lattice, along with the localized portion of the energy, there is always a propagating part due to the existence of extended modes at low frequencies. A quantitative description of wave propagation in disordered $1$D systems of one degree of freedom (DoF) per lattice site was formulated in Refs.~\\cite{Ishii, Kundu, Lepri} where wave-packet spreading was quantified using both analytical and numerical methods. Moreover, many variations of these $1$D lattices have been studied extensively in several works including dynamical regimes ranging from the homogeneous linear to the disordered nonlinear~\\cite{Chiaradis,Guebelle,MasonPanos,vassospre2016,Luding2,Sen2017poly,jk,arn_gran,many2}. \n\nAs a natural extension to the above studies, an investigation into the corresponding behavior in disordered lattices with more than one DoF per site seems plausible. Not many studies along such lines have been reported in the literature. The majority of existing works have taken the approach of making generalizations of the tight binding model by assuming a linear coupling between two (or more) $1$D chains~\\cite{ladderPRB,flachlieb}. Such coupling results in changes to the dispersion relation thereby changing the energy transport properties. Our recent work with a linear disordered phononic lattice \\cite{arn_phononic} is indeed an attempt to fill this gap. The wave dynamics of disordered harmonic chains with two DoFs per site appear to be quite interesting, deserving further investigations. Such systems have been useful in modeling macroscopic mechanical devices including granular chains, highly deformable elastic assemblies and origami lattices~\\cite{pichard,florian1,yasuda,yasuda2,deng2,florian2}. This allows for easy  tunability of the system's dispersion due to the geometrical characteristics and material properties, and makes these systems attractive for several applications.\n \nHere we focus on highly deformable architected lattices, characterized by a nonlinear response which enabled the design of new classes of tunable and responsive elastic materials. Several such soft structures have already been reported including bioinspired  soft robots \\cite{shepherd, yang}, self-regulating microfluidics \\cite{Beebe}, reusable energy absorbing systems \\cite{shan,restrepo}, materials with programmable response \\cite{florijn} and information processing via physical soft\nbodies \\cite{nakajima}. Furthermore, soft architected materials present opportunities to control the propagation of elastic waves, since their dispersion properties can be altered by applying a large, nonlinear pre-deformation \\cite{boyce1,boyce2,casadei} or changing the geometry \\cite{deng2}. To date, most of the investigations have predominantly focused on linear stress waves, or soliton solutions of such systems due to the capability of the soft structures to support large-amplitude nonlinear waves. Here, taking a step forward, we study a particular lattice  that supports both translational and rotational waves~\\cite{deng2}. Our main goal is to understand how nonlinear lattice waves propagate in the presence of strong disorder when the DoFs are coupled, as well as the system's chaoticity.\n \nThe rest of this paper is arranged as follows: In Section~\\ref{sec2} we describe the Hamiltonian model of the lattice structure and also formulate the system's equations of motion. The dispersion relation of the system, in addition to its dynamics in the linear limit is also discussed. In Section~\\ref{sec3} we investigate the nonlinear effects on wave propagation under strong disorder, as well as study the system's chaoticity and finally, in Sec.~\\ref{sec4}, we summarize our findings and present our conclusions.\n\\section{The Hamiltonian model and its equations of motion} \\label{sec2}\n\nThe $1$D elastic mechanical lattice studied in this work is assembled from an array of aligned LEGO\\textsuperscript{\\textregistered} crosses connected by flexible links \\cite{celli} as depicted in Fig.~\\ref{fig1}. This system constitutes a highly deformable elastic lattice supporting both translational and rotational waves ($2$ DoFs per site). In Ref.~\\cite{deng2}, the authors describe the general equations of motion for a structure that takes into consideration \nsome of the possible geometrical variations of the lattice. However, for the purposes of this work, we limit ourselves to an aligned, symmetrical structure. The crosses are joined to their neighbors by some flexible hinges which are modeled using a combination of three linear springs. The stretching is modeled by a spring with stiffness $ k_{l}$\nand the shearing is described by a spring $ k_{s}$, whilst the bending is modeled by a torsional spring $ k_{\\theta}$ [see Fig.~\\ref{fig1} (b)].\n\nBy making use of the spatial periodicity $a$, we recast the horizontal deflections $u_n$ at the $n$th lattice site to $U_n = u_n\/a$ and change time units to dimensionless time, $ T = t\\sqrt{ k_{l}\/\\overline{m} } $ and the springs are normalized as $K^{(\\theta)} = 4 k_{ \\theta }\/k_{l} a^2$ and $K^{(s)} = k_{s}\/k_{l}$. The mass of each unit of the cross $m_n$ is normalized such that $M_n = m_n\/\\overline{m}$ where $\\overline{m} $ is the average mass of the crosses. This implies that in the case of the homogeneous chain, $M=M_n  = m_n\/\\overline{m} = 1$ and $\\Gamma=\\Gamma_n  = J_n\/ \\overline{m} a^2$. Herein, $J_n$ denotes the rotational moment of inertia at the $n$th lattice site.\nFor the rest of this work, we assume the parameters of the experimental setup of Ref.~\\cite{deng2} where each cross has mass $m = 4.52$~g ($\\overline{m}=m$), $a = 42$~mm and $J = 605$ g$\\cdot$mm$^2$.\nAlso the spring constants are $k_{l}=71.69$ N$\\cdot$mm$^{-1}$, $ k_{s} = 1.325 $N$\\cdot$mm$^{-1}$ and $k_{\\theta} = 4.85$ N$\\cdot$mm.\n\nThe Hamiltonian $H$, of the top or bottom layer of the system is thus given as (see Ref.~\\cite{deng2} for details)\n\\begin{figure}[t!] \n\\centering\n\\includegraphics[width=12.20cm]{lego_schema.png} \\\\\n\\includegraphics[width=12.20cm]{lego_deflections.png}\n\\caption{\\label{fig1} (a) An architected, highly deformable and elastic mechanical structure which supports translational and rotational waves. (b) Schematic of the cross pairs [by symmetry, the dynamics can be described by either the top row or bottom row of (a)] showing the translational and angular deflections deflections. The connectors (marked in red) model a combination of bending $k_{\\theta}$, shear $k_s$ and stretching springs $k_l$.}\n\\end{figure}\n\\begin{equation}\nH  =\\sum_{n=1}^{N} \\left\\{  \\frac{ M_n \\dot{U}^2_n}{2} +   \\frac{ \\Gamma_n \\dot{\\theta}^2_n }{2} +  \\frac{1}{2}  \\Delta^{n~^{2}}_{_{LH}} + \\frac{K^{(s)}}{2}  \\Delta^{n~^{2}}_{_{SH}}   + \\frac{K^{(\\theta)} }{8 } \\left(  \\delta^{n~^{2}}_{_{\\theta H}}  + \\frac{1}{2}  \\delta^{n~^{2}}_{_{\\theta V}} \\right) \\right\\},\n\\label{hamil} \n\\end{equation}\nwhere the dimensionless deflections are given by, \\\\\n$ \\Delta^{n}_{_{LH}}  = U_{n+1} - U_{n} + \\frac{1}{2} \\left( 2 - \\cos{ \\theta_{n}}  -  \\cos{ \\theta_{n+1}} \\right),$\n$ \\Delta^{n}_{_{SH}}  = \\frac{1}{2} \\left( \\sin{ \\theta_{n+1}}  -  \\sin{ \\theta_{n}} \\right),$\n$  \\delta^{n}_{_{\\theta H}} = \\theta_{n+1} + \\theta_{n}, $ and   $\\delta^{n}_{_{\\theta V}} = 2 \\theta_{n}.$\nIn Eq.~(\\ref{hamil}), $[~\\dot{}~]$ denotes the derivative with respect to time\n\nWe derive the equations of motion from the Hamiltonian Eq.~(\\ref{hamil}) which yields\n\\begin{equation}\n\\begin{aligned}\n M_n  \\ddot{U}_{n}  & =  \\Big[ U_{n+1} - U_{n} + \\frac {1}{2} \\{  2 -  \\cos( \\theta_n)  -  \\cos( \\theta_{n+1}) \\}  \\Big] \\\\\n   - &\\Big[ U_{n} - U_{n-1} + \\frac {1}{2} \\{ 2 -  \\cos(\\theta_n)  -  \\cos( \\theta_{n-1}) \\}  \\Big ],\n\\label{em1}\n\\end{aligned}\n\\end{equation}\n\n \\begin{equation}\n\\begin{aligned}\n \\Gamma_n \\ddot{\\theta}_{n} & =   \\frac{1}{4} K^{(s)} \\cos( \\theta_n)  \\Big[ \\sin( \\theta_{n+1})  -  \\sin( \\theta_{n})   \\Big]  \\\\\n  + &\\frac{1}{4} K^{(s)}  \\cos( \\theta_n)  \\Big[ \\sin( \\theta_{n-1})  -  \\sin( \\theta_{n})   \\Big]  \\\\\n+ &\\frac{1}{4} \\sin( \\theta_n)  \\Big[ 2 (U_{n} - U_{n+1}) + \\cos( \\theta_n)  + \\cos( \\theta_{n+1}) - 2  \\Big] \\\\\n + &\\frac{1}{4} \\sin( \\theta_n)  \\Big[ 2 (U_{n-1} - U_{n}) + \\cos( \\theta_n)  + \\cos( \\theta_{n-1}) - 2  \\Big]  \\\\\n-&\\frac{1}{4} K^{(\\theta)} ( \\theta_{n+1} + 4\\theta_{n} + \\theta_{n-1} ).\n\\label{em2}\n\\end{aligned}\n\\end{equation}\n\n\\subsection{  Homogeneous linear system} \\label{ssec1}\nFirst let us consider the homogeneous system\nand linearize the nonlinear terms (trigonometric terms) in \nEqs.~(\\ref{em1}) and (\\ref{em2}) by assuming small angles, $\\theta_{n+p}$ with $p = \\left\\{-1,0,1\\right\\} $,\nand taking a power series expansion of the appropriate cosine and sine terms to give the first two lowest order terms as\n\n\\begin{equation}\n \\sin \\theta_{n+p} \\approx  \\theta_{n+p} - \\frac{1}{6} \\theta_{n+p} ^3 + \\ldots,\n\\label{eq2}\n\\end{equation}\n\n\\begin{equation}\n \\cos \\theta_{n+p} \\approx  1 - \\frac{1}{2} \\theta_{n+p} ^2 + \\ldots.\n\\label{eq3}\n\\end{equation}\nThe linear parts of Eqs.~(\\ref{eq2}) and (\\ref{eq3}) are plugged appropriately into \nEqs.~(\\ref{em1}) and (\\ref{em2}) to give the linear equations of motion as\n\\begin{equation}\n\\begin{aligned}\n  M_n \\ddot{U}_{n}  & =  U_{n+1} - 2 U_{n} + U_{n-1}  ,\n\\label{em3}\n\\end{aligned}\n\\end{equation}\nand \n \\begin{equation}\n\\begin{aligned}\n \\Gamma_n \\ddot{\\theta}_{n} & =  \\tilde{K} (\\theta_{n+1} -2\\theta_{n} + \\theta_{n-1}) - 6 K^{(\\theta)} \\theta_{n},\n\\label{em4}\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{K} = K^{(s)} - K^{(\\theta)}$.\nFor the homogeneous case, $M_n = M = 1$ and $\\Gamma_n = \\Gamma$ as already defined. \nA quick glance at Eqs.~(\\ref{em3}) and (\\ref{em4}) reveals that the two sets of DoFs are decoupled in the linear regime. Eq.~(\\ref{em3}) belongs to the linear FPUT class of equations and Eq.~(\\ref{em4}) is a linear KG-type equation.\n\nWe now consider solutions of the form\n\\begin{equation}  \n\\mathbf{X}_n =  \\begin{pmatrix} \n            U_n (t)\\\\\n            \\theta_n (t)\n           \\end{pmatrix} = \\mathbf{X}e ^{i \\omega t - i q n},\n \\label{eq4}          \n\\end{equation}\n where $\\mathbf{X}=[U_0,\\Theta_0]$ is the amplitude vector, $\\omega$ is the cyclic frequency\nand $q$ is the wave number. Inserting Eq.~(\\ref{eq4}) into Eqs.~(\\ref{em3}) and ~(\\ref{em4}), we obtain the following eigenvalue problem for the allowed frequencies \n$\\mathbf{D} \\mathbf{X} = \\Omega^2 \\mathbf{X}$,\nwhere the resultant dynamical matrix is\n$$ \\mathbf{D} = \\begin{pmatrix}\n     2 -2 \\cos q  & 0 \\\\\n      0 &  \\frac{1}{2 \\Gamma}\\left [(K^{(s)} + 2 K^{(\\theta)}) - ( K^{(s)} - K^{(\\theta)}) \\cos q \\right ]  \n         \\end{pmatrix}.\n         $$\nThe corresponding dispersion branch for the transverse DoFs is\n\\begin{equation}\n \\omega^{(U)} = \\sqrt{2 -2 \\cos q},\n\\label{eq5}\n\\end{equation}\nwhilst for the rotational DoFs,\n\\begin{equation}\n \\omega^{(\\theta)} = \\frac{1}{\\sqrt{4 \\Gamma}} \\sqrt{2\\left (K^{(s)} + 2 K^{(\\theta)}) - 2( K^{(s)} - K^{(\\theta)}\\right ) \\cos q  }.\n\\label{eq6}\n\\end{equation}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=12.50cm]{disp.png}\n\\caption{\\label{fig2} The two dispersion branches of the system for translational ($\\omega^{(U)}$ - red curve) and rotational ($ \\omega^{(\\theta)}$ - blue curve) DoFs [See respectively Eqs.~(\\ref{eq5}) and (\\ref{eq6})].}\n\\end{figure}\nThe dispersion relation for the two different DoFs as given by Eqs.~(\\ref{eq5}) and (\\ref{eq6}), are plotted in Fig.~\\ref{fig2}.\nAn important feature for the system is that the rotational mode branch (blue curve in Fig.~\\ref{fig2}) starts at a finite frequency. This means that linear rotational waves are not supported for $\\omega < \\frac{1}{\\sqrt{4 \\Gamma}} \\sqrt{6 K^{(\\theta)}}$ since this frequency domain corresponds to a band gap. In fact, the linear dispersion relation of the rotations can be directly mapped onto the one for the linear KG lattice. \nOn the other hand the transverse displacements follow a typical linear mass-spring dispersion relation.\n\n\\subsection{ Disordered linear system} \\label{ss2b}\nIn order to consider to consider a disordered version of the system, we first note that the disordered linear KG system exhibits AL (See Ref.~\\cite{Kundu} for the corresponding behavior of the linear FPUT). Additionally, even though there are a number of ways to introduce disorder in the system, we model the crosses of the architected lattice assuming disorder in the masses $M_n$, which in turn implies disorder in the rotational moments of inertia $\\Gamma_n$. In practice, disorder in the system can also be achieved by changing the material used to manufacture the LEGO\\textsuperscript{\\textregistered} bricks at each site \\cite{celli} without changing their geometrical dimensions. In dimensionless units, the masses are normalized to unity for a homogeneous chain hence we take this into consideration when choosing the disorder distribution and take $M_n$ from a uniform probability distribution $ f \\big( M_n \\big) $ where\n\\[\n f \\big( M_n \\big)  =\n  \\begin{cases}\n                                   W^{-1}, &-W\/2    < \\text{ $ M_n $} - 1 < W\/2, \\\\\n                                   0 & \\text{otherwise}. \\\\\n    \\end{cases}\n\\]\n$ W$ denotes the distribution width and for this study, we choose $ W = 1.8$ hence $ 0.1 \\leq M_n \\leq 1.9 $. \n\nTo study the dynamical behavior of the system, the equations of motion are integrated using the \\texttt{ABA864} symplectic integrator \\cite{blanes} which has been proved to be very efficient for the accurate integration of large Hamiltonian lattice models \\cite{senyange2018,danieli2019}.\nThis integration scheme allows for energy conservation of the total energy $H$, and keeps the relative energy error $\\Delta H (T)  = \\left|\\dfrac { H (T) - H (0)  }{ H (0) }\\right| <  10^{-5}$ \nwhen the integration time step is set to be $ \\tau = 0.1$. In all our numerical simulations, we employ fixed boundary conditions i.e.,~$ U_{0} = U_{N+1} = 0$, $ \\theta_{0} = \\theta_{N+1} = 0$ ,  $ \\dot{U}_{0} = \\dot{U}_{N+1} = 0$ and $ \\dot{\\theta}_{0} = \\dot{\\theta}_{N+1} = 0$. Furthermore, the considered lattice size is large enough so that the energy does not reach the lattice boundaries. A typical numerical integration of the nonlinear system for $ T = 10^5$ requires a lattice size of at least $2 \\times 10^5$ sites.\n\n\\subsubsection{Translational degrees of freedom}\nWe start our analysis by first exploring the two possible single site initial excitations of velocity and displacement for the translation DoFs i.e.,~\n\n\\begin{equation}\n\\dot{U}_{N\/2}(0)=\\sqrt{2H\/M_{N\/2}}, \\quad \\textrm{or} \\quad U_{N\/2}(0)= \\nu,\n\\label{eq7} \n\\end{equation}\nindependently. In Eq.~(\\ref{eq7}), the scalar $\\nu$ is real and its value is altered to match the desired system energy. We fix the total system energy at $H=H_{0}$ and integrate the system up to $T = 10^5$ time units and observe how the initially localized wave-packet evolves in time. A typical scenario of the dynamics is shown in Figs.~\\ref{fig3}(a)-(b) for $H_0 = 10^{-4}$. For single site velocity initial excitations, the energy distribution displays two main parts, a central localized part, and an expanding peripheral part which is spreading beyond the excitation point [see Fig.~\\ref{fig3}(a)]. Similar spreading characteristics are observed in the dynamics for displacement initial excitations as depicted in Fig.~\\ref{fig3}(b).\n\nFor a more quantitative description, we follow the time evolution of the participation number $P$ and the second moment $m_2$, of the energy distribution to characterize the localization and spreading properties of the wave-packet. The former quantity $P$, is used to quantify the number of highly excited sites in a lattice~\\cite{flachbook} and is computed as\n\\begin{equation} \\label{eqp}\nP = 1\/ \\sum h_n^2,\n\\end{equation}\nwhere $h_n = H_n \/ H$ is the normalization of the site energy $H_n$. In the case of equipartition, for a lattice of size $N$, $ P = N$, while the other extreme gives $ P = 1$, for a wave-packet in which only a single site is highly excited.\nThe second moment $ m_2 $~\\cite{Kundu} of the energy distribution given by\n\\begin{equation} \\label{eqm2}\nm_{2} =  \\sum_n (n-\\overline{n})^2 H_n \/H,\n\\end{equation}\nis a measure of the wave-packet's extent where $\\overline{n} = \\sum_n n H_n\/H$ is the mean position of the energy distribution.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=6.20cm]{Fput_MLin.png}\n\\includegraphics[width=6.20cm]{Fput_DLin.png}\\\\\n\\includegraphics[width=6.20cm]{p_fputlin.png}\n\\includegraphics[width=6.20cm]{b_fputlin.png}\n\\caption{\\label{fig3} Spatiotemporal evolution of the energy distribution for a representative realization with (a) velocity and (b) displacement single site initial excitation in the linear system. The colorbars in (a) and (b) are in log-scale. (c) Time evolution of the average participation number $\\langle P \\rangle$, (d) estimation of the exponent $\\beta$, related to the time evolution of the average second moment through $ \\langle m_2 \\rangle \\propto T^{\\beta} $. For velocity (black curves) and displacement (magenta curves) initial excitations, the mean values $ \\langle \\cdot \\rangle$ are calculated from $100$ disorder realizations and the shaded areas represent the statistical error (one standard deviation). The dashed lines in (d) indicate $\\beta =1.5$ (top) and $\\beta =0.5$ (bottom) and the system energy for all realizations is $ H = 10^{-4} $. }\n\\end{figure}\n\nThe time evolution of $\\langle P \\rangle$ indicates that a  maximum value is reached and remains constant for both velocity and displacement excitations as illustrated in Fig.~\\ref{fig3}(c) by the black and magenta curves respectively. In this work, $ \\langle \\cdot \\rangle$ denotes averages over $100$ disorder realizations. One of the differences between the two cases is that velocity initial excitations yield slightly higher values of $\\langle P \\rangle$ when compared to the $\\langle P \\rangle$ reached for displacement initial excitations. This is due to the fact that more low frequency propagating modes are excited for the case of velocity initial excitation than with displacement initial excitation \\cite{Kundu}. \nThe time evolution of $\\langle m_ 2 \\rangle$ for the two excitations is also different for the same reason.\n \n Regarding the second moment $m_2$, the usual practice is to assume that $\\langle m_2 \\rangle  \\propto T^{\\beta}$. Then the parameter $\\beta$ is numerically estimated by the time local derivative\n\\begin{equation} \\label{eqb}\n\\beta=  \\frac{ d  \\log_{10}\\langle m_2 (T) \\rangle }{ d  \\log_{10} T}.\n\\end{equation}\nThe exponent $\\beta$ is used to quantify the asymptotic behavior of $\\langle m_2 \\rangle$ for sufficiently large times. It is calculated by first smoothing the $m_2 (T)$ values of each disorder realization through a locally weighted difference algorithm~\\cite{cleveland1,cleveland2} and then averaged over all realizations. The computed $\\beta$ for the two cases, saturates to $\\beta = 1.5$ and $\\beta = 0.5$, for respectively velocity and displacement single site excitations as shown in Fig.~\\ref{fig3}(d). This is an expected result because we have already shown the system to be practically a linear disordered FPUT lattice, since the translational DoFs do not couple to the rotational DoFs. \nHere we note that, according to the full system of equations [Eqs.~(\\ref{em1})-(\\ref{em2})] single site translational excitations as given by Eq.~(\\ref{eq7}) will never couple the two DoFs. This is a particularity of the structural geometry under consideration. Thus,  whatever the initial excitation energy, single site initial translations will always lead to the linear behavior summarised in Fig.~\\ref{fig3}. For this reason we shall not consider such initial conditions in Section~\\ref{sec3}. \n\n\\subsubsection{ Rotational degrees of freedom} \\label{sssecR}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=6.20cm]{KG_MLin.png}\n\\includegraphics[width=6.20cm]{KG_DLin.png}\\\\\n\\includegraphics[width=6.20cm]{kg_e4_P.png}\n\\includegraphics[width=6.20cm]{kg_e4_B.png}\n\\caption{\\label{fig4} Similar to Fig.~\\ref{fig3} but for initial angular deflections (magenta curves) and angular deflection time derivatives (black curves). The vertical axes for (a) and (b) are given in $\\log_{10} $. The dashed line in (d) indicates $\\beta=0$. }\n\\end{figure}\n\nWe now turn our attention to the dynamics of single site rotational excitations (angular deflection or the angular deflection time derivative). These excitations are expected to follow the dynamics of a discrete linear KG lattice as already explained earlier in Section.~\\ref{ssec1}. More specifically, we consider independently the initial excitations\n\\begin{equation}\n\\dot{\\theta}_{N\/2}(0)=\\sqrt{2H\/\\Gamma_{N\/2}},\\quad \\textrm{or} \\quad \\theta_{N\/2}(0)= \\mu,\n\\label{eq72} \n\\end{equation}\nwhere $\\mu$ is real and is chosen to match the desired system energy $H=H_0$.\n\nThe energy distribution profiles for both angular deflection time derivative (black curves) and angular deflection (magenta curves) excitations [see Figs.~\\ref{fig4}(a) and (b)] show localized energy distributions for times $T \\gtrsim 10^4$ after an initial phase of wave-packet spreading.\nThe subtle differences between angular deflection time derivative and angular deflection initial excitations in panels (a)-(b) of Fig.~\\ref{fig4} are mainly due to the differences in the initially excited modes of the system for each respective excitation. Thus, the dynamics of the rotations in the linear limit show complete Localization in sharp contrast to the perpetual wave-packet spreading observed for the translational DoFs [Figs.~\\ref{fig3}(a) and (b)]. This is a consequence of the fact that the KG system Eq~(\\ref{em4}) describing rotations, is known to map to the DNLS and experiences localization of all modes \\cite{kivshar,kivshar1993,johansson2006}. To quantify the localization for the initial angular deflections (magenta curves) and angular deflection time derivatives (black curves) we plot the time evolution of $ \\langle P \\rangle $ which reaches constant finite values after $ T \\gtrsim 4 \\times 10^{4} $ [Fig.~\\ref{fig4}(c). A similar behavior is observed for $ \\langle m_2 \\rangle $ (not shown here) for  $ T \\gtrsim 10^{3} $ ] in agreement to what is expected for a linear disordered KG chain. The final saturation of the $ \\langle m_2 \\rangle $ value is clearly depicted in the evolution of the exponent $\\beta$ from the relation $ \\langle m_2 \\rangle \\propto T^{\\beta} $, which eventually becomes $ \\beta =0 $ as clearly seen in Fig.~\\ref{fig4}(d), showing that indeed the system is effectively a $1$D linear disordered KG chain.\n\n\\section{ Disordered nonlinear system} \\label{sec3} \n\nHaving considered the behavior of the system in the linear limit, we further study the fully nonlinear system as described by Eqs.~(\\ref{em1})-(\\ref{em2}). Before proceeding further, we reiterate that single site initial translational velocity ($\\dot{U}_{N\/2}$) or displacement ($U_{N\/2}$) excitations \\textit{do not induce a nonlinear response}. However, initial rotational excitations induce both a nonlinear response on rotations as well as nonlinear coupling between the two sets of DoFs. For the rest of this work we focus exclusively on single site (in the centre of the lattice) initial conditions of angular deflections as well as angular deflection time derivatives. \n\n\\subsection{Weakly nonlinear regime} \n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=14.0cm]{fig5new.png}\n\\end{centering}\n\\caption{\\label{fig5} Weakly nonlinear system,~ (a)-(d) Time evolution of $ \\langle P \\rangle $ (\\ref{eqp}), $\\langle m_2 \\rangle$ (\\ref{eqm2}), $\\beta$ (\\ref{eqb}) and $\\langle \\Lambda \\rangle$ (\\ref{eq41}), respectively. The black (magenta) curves correspond to single site initial angular deflection time derivatives (angular deflection) excitations. The average values $\\langle \\cdot \\rangle$ are computed from $100$ disorder realizations and the lightly shaded areas represent the statistical error (one standard deviation). The dashed blue and green curves which are not always visible due to their overlapping with other curves, respectively show results of the linearized system for the initial conditions given by Eq.~(\\ref{eq72}). All results are for the weakly nonlinear regime with $H = 10^{-8}$. The horizontal line in (c) indicates $\\beta =0$. }\n\\end{figure}\n\nWe start by implementing single site angular deflections and time derivatives of angular deflections as initial excitations for a sufficiently small energy ($ H = 10^{-8}$) so that the system is in the weakly nonlinear regime. With the physical system at hand, this energy corresponds to an initial angle deflections of $\\approx 0.2^{\\circ}$. The time evolution of $\\langle P \\rangle$ is shown in Fig.~\\ref{fig5}(a) where we observe a saturation to a constant value for each type of excitation [angular deflection time derivative (black curve) and angular deflection (magenta curve)]. In fact we also plot, in the same figure, the corresponding linear result (dashed curves) and we observe that the weak nonlinearity does not affect the participation number. Nevertheless, we find that nonlinearity plays a significant role on the evolution of $\\langle m_2 \\rangle$ and $\\beta$. This is illustrated in Figs.~\\ref{fig5}(b)-(c) where both these quantities are found to increase for the last two decades ($T \\gtrsim 10^3$) of the evolution clearly indicating energy spreading  in the system.\n\nTo further explore the spreading it is worthwhile to also investigate the chaoticity of the system using the finite time maximum Lyapunov exponent (ftMLE). It is often found, in systems with multiple degrees of freedom, that energy spreading is due to chaos around the excitation region \\cite{disorder_15,skokos2013,Bob2018}. The ftMLE,\n\\begin{equation} \\label{eq41}\n\\Lambda(T) = \\frac{1}{T} \\ln \\frac { || \\mathbf{w} (T) || }{ || \\mathbf{w} (0) ||},\n\\end{equation}\nis computed using the so-called standard method \\cite{benettin,lec_notes}. $\\mathbf{w} (T)$ is a vector of small perturbations from the phase space trajectory at time $T$ (also called deviation vector) which we denote as \n\n\\begin{equation}\n\\begin{aligned}\n \\mathbf{w} (T)  = & [ \\delta  U_1 (T), \\ldots , \\delta  U_N (T), \\\\ & \\delta  \\theta_1 (T), \\ldots , \\delta  \\theta_N (T), \\\\\n &M_1 \\delta \\dot{U}_1 (T),  \\ldots , M_N \\delta \\dot{U}_N (T), \\\\\n &\\Gamma_1 \\delta  \\dot{\\theta}_1 (T),  \\ldots , \\Gamma_N \\delta \\dot{\\theta}_N (T)], \n \\end{aligned}\n\\end{equation}\nwhere $\\delta  U_n (T)$ and $\\delta  \\theta_n (T)$ indicates small perturbations in positions, while $M_n \\delta \\dot{U}_n (T)$ and $\\Gamma_n \\delta \\dot{\\theta}_n (T)$ indicate small perturbations in momenta for the two DoFs at the $n$th lattice site. The mLE is defined as $ \\lambda = \\lim_{T \\to \\infty} \\Lambda (T) $. In Eq.~(\\ref{eq41}), $|| \\cdot ||$ denotes the usual Euclidean vector norm. For chaotic trajectories, $\\Lambda $ attains a finite positive value, otherwise, $\\Lambda \\propto T^{-1}$ for regular orbits. An efficient and accurate method to follow the evolution of $\\mathbf{w} (T)$ is to numerically integrate the so-called variational equations \\cite{galgani}, which govern the vector's dynamics, together with the Hamilton equations of motion using the tangent map method outlined in Refs.~\\cite{gerlach1,gerlach2,gerlach3}. The mLE can be used to discriminate between regular and chaotic motions since $\\Lambda  = 0$ for regular orbits and $\\Lambda > 0$ for chaotic orbits. The magnitude of the mLE can also be used as a measure of the chaoticity: larger mLE values imply stronger chaotic behaviors.\n\n In Fig.~\\ref{fig5}(d) we show the calculated ftmLE which is found to follow the power law decay $ \\langle \\Lambda \\rangle \\propto T^{-1}$,  and thus the system exhibits regular dynamics. As such we conclude that  the observed energy spreading \\textit{cannot} be attributed to chaoticity as is the case for other lattice models including the single DoF per site KG lattice \\cite{skokos2013,Bob2018}.\n\nIn order to explain the spreading we need to further monitor the energy density in the lattice as a function of time. In Fig.~\\ref{fig6}(a) we show four snapshots of the energy density around the initial excitation point ($n=0$). The spatial distribution of energy is separated into two distinct regions: a large amount of energy localized around  $n=0$  and an extended tail with much lower energy (five orders of magnitude less). In fact, if we consider the propagation of the higher energy central part, especially for $T \\lesssim 10^3$, we clearly observe a leading wave-front (dashed vertical lines) which propagates slower than the main leading wave-front. The latter corresponds to the low energy regions [$H_n \\lesssim 10^{-13}$ in Fig.~\\ref{fig6}(a)] which propagates with a normalized velocity close to one. On the contrary the high energy part around the center [$H_n \\gtrsim 10^{-13}$] is propagating much slower with a normalized velocity of $\\approx 0.1$. These two velocities correspond respectively to the largest group velocities of the translation and rotational branches of the dispersion relations shown in Fig.~\\ref{fig2}. Thus we conjecture that the two distinct parts of the energy distribution, high and low, correspond respectively to the two different types of DoFs i.e., the rotational and translation deflections.\n\nFurthermore, as shown by the snapshots for $T\\gtrsim 10^4$ in Fig.~\\ref{fig6}(a), at later times only the lower (translation) part of the energy continues to spread.\nThis fact is corroborated by calculating the exponent of the second moment for energies lower (larger) than a threshold ($H=10^{-13}$) as shown in Fig.\\ref{fig6}(b). The exponent $\\beta$ for the high energy central part almost vanishes (red curve), indicating no spreading while for the low energy region value of $\\beta$ (black curve) is finite revealing spreading. Thus we conclude this sub-section by explaining the dynamics in the following manner. In the weakly nonlinear regime, the role of the nonlinearity is to induce spreading by stimulating the translation DoFs (which have an FPUT character) through the nonlinear coupling. In this way, the energy spreading in this regime is characterized by a hybrid of KG-like and FPUT-like behaviors. In addition the rate of wave-packet spreading as quantified by $\\beta$ shows no defined asymptotic behavior but rather a dependency on time [Fig. \\ref{fig5}(c)]. This is in accordance with the results obtained for a weakly nonlinear FPUT lattice by Lepri \\textit{et.al.}, \\cite{Lepri}.\n \n\\begin{figure}\n\\centering\n\\includegraphics[width=12.0cm]{fig6newa.png}\\\\\n\\includegraphics[width=12.0cm]{fig6newb.png}\n\\caption{\\label{fig6} (a) Average energy distribution profiles of $100$ disorder realizations for $T= 5\\times 10^2$~(red curve) , $T= 10^3$~(blue curve), $T= 10^4$~(magenta curve) and $T= 5\\times 10^4$~(black curve). The dotted horizontal line marks the energy $ \\langle H_n \\rangle = 10^{-13}$ while the vertical dashed lines indicate the position of the slower wave-front at the indicated times. (b) Time evolution of the exponents $\\beta$ for the central part (solid red curve) and the tails (solid black curve). The lightly shaded regions in (b) indicate the statistical error (one standard deviation). All panels are for angular deflections of system energy $H=10^{-8}$.}\n\\end{figure}\n\n\\subsection{Strongly nonlinear regime} \nLet us move away from the weakly nonlinear regime and increase the nonlinearity of the system by increasing the initial excitation energy to $ H = 5 \\times 10^{-3}$. This energy corresponds to large initial angle deflections of about $30^{\\circ}$. Note that this value of energy leads to strong nonlinear behavior. The system, in this regime, shows a completely different behavior of spreading as is indicated by the increase of $\\langle P \\rangle$ during the time evolution as shown in Fig.~\\ref{fig7}(a). The number of highly excited sites grows in time contrary to what we observed for the weakly nonlinear regime, which shows practically no growth in $\\langle P \\rangle$ at long times. As expected, this increase of participating particles leads to wave-packet spreading and this is confirmed by the time evolution of $\\langle m_2 \\rangle$ which is also increasing as indicated in Fig.~\\ref{fig7}(b). A feature we observe in this strong nonlinear regime is that the dynamics no longer depends on the type of initial condition (angular deflections or time derivatives of angular deflections). This is noticeable from the practically overlapping black (time derivatives of angular deflections) and magenta (angular deflections) curves in Figs.~\\ref{fig7}(b)-(c). To quantify wave-packet spreading, we estimate the exponent of $\\langle m_2 \\rangle \\propto T^{\\beta}$ which is found to acquire values around $\\beta \\approx 2$ as shown in Fig.~\\ref{fig7}(c). This value indicates a very strong spreading corresponding to a near ballistic propagation. We find this result to be quite interesting since the nonlinearity of the flexible architected material under study, is strong enough to bring the system to ballistic behavior, which is not always the case in other systems such as the FPUT and KG lattices~\\cite{pikovsky,flachbook}. In fact, the other example to our knowledge, where near ballistic behavior in a disordered system is observed, is for mechanical lattices featuring non-smooth nonlinearities due to Hertzian forces \\cite{jk}. \n \n Also, in this strongly nonlinear regime the distinct behavior of the two types of DoFs that was observed in Fig.~\\ref{fig6}(a) is now lost. According to Fig.~\\ref{fig7}(d) showing the mean profile of the energy distribution, we identify a large part of the energy being localized around the center and a propagating tail  travelling almost ballistically. This is expected since according to Eqs.~(\\ref{em1})-(\\ref{em2}) the coupling of the two types of DoFs is enhanced at each lattice site when the rotations are of high amplitude. Thus we no longer distinguish between a KG-like and FPUT-like evolution of the energy profiles. We have also considered a range of system energies between $H=10^{-8} - 10^{-3}$ and found that the distinction between KG- and FPUT-like behaviors gradually disappears as the system energy is increased. Some of the results for the intermediate energies  are reported in Ref.~\\cite{ngapasarephd}.\n \nRegarding the chaoticity of the system, for such high initial angles and thus strong nonlinearity, we find the dynamics to be chaotic. This is evident in Fig.~\\ref{fig8}(a) where the mean value of the ftMLE $\\Lambda$ Eq.~(\\ref{eq41}), is shown to be decreasing in a much slower rate compared to regular dynamics (dashed line). This type of chaotic behavior, where the ftMLE does not reach an asymptotic constant value, has recently attracted much attention and appears to be a particular case of chaos spreading \\cite{disorder_15, Bob2018, many2} and it is related to the fact that as the wave-packet spreads, the constant total energy is shared among more sites and consequently the energy per excited site (which plays the role of active nonlinearity strength) decreases. Consequently, the ftMLE which is a global measure of chaos, is also decreasing. \n\nHowever, different to what was found in Refs.~\\cite{disorder_15, Bob2018, many2}, for the architected lattice under study, here the slope of $ \\langle \\Lambda \\rangle$ does not reach a constant value slope even for the \\textit{largest possible } angular deflection  of $45^{\\circ}$.\n %\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12.60cm]{fig1000.png}\n\\end{center}\n\\caption{\\label{fig7} (a)-(c) Similar to Figs.\\ref{fig5}(a)-(c). (d) Average energy density profiles over $100$ disorder realizations at $T=10^2$ (blue curve), $T=5 \\times 10^2$ (red curve) and $T=10^3$ (black curve). Results in all panels are for energy $H= 5 \\times 10^{-3}$.}\n\\end{figure}\n %\nThus, for the sake of completeness and to be able to compare the results regarding the chaos spreading of the soft architected lattice with other models in the literature, we extend our numerical simulations using even larger initial energies. In particular, in Figs.~\\ref{fig8}(b)-(c) we show results using an initial energy of $H=10^{-1}$.\nNote that this value of energy leads to an even stronger nonlinear behavior than the penultimate case above. \n %\n\n %\n %\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12.5cm]{fig8feb.png}\n\\end{center}\n\\caption{\\label{fig8}~(a) Time evolution of the average ftMLE, Eq.~(\\ref{eq41}), $ \\langle \\Lambda \\rangle $ for the strongly nonlinear regime angular deflections and time derivatives of angular deflections as initial excitations for energy $H=5 \\times 10^{-3}$. The red dashed line indicates the power law $\\langle \\Lambda \\rangle \\propto T^{-1}$. (b) Similar to (a) but for energy $H=10^{-1}$. Here the red dashed line indicates the power law $\\langle \\Lambda \\rangle \\propto T^{\\kappa}$, with $\\kappa = -0.28 $. (c) Time evolution of the exponent $ \\kappa $ at energy $H = 10^{-1}$. In all panels the magenta and black curves respectively show results for angular deflections and time derivatives of angular deflections as initial excitations. The magenta horizontal dashed line indicates $\\kappa = -0.28$. }\n\\end{figure}\nWe also observe in Fig.~\\ref{fig8}(b) that the ftMLE reaches a clearly constant slope with which it is decaying. In particular, the slope of the exponent assuming that $\\langle \\Lambda \\rangle \\propto T^{\\kappa}$  is found to be approximately $\\kappa = -0.28$ for both angular deflections and time derivatives of angular deflections. Note that this asymptotic value of the exponent $\\kappa$, is comparable to the corresponding values for the ftMLE in the more studied cases of $1$D lattices namely the disordered DNLS equation and the disordered KG model \\cite{skokos2013, Bob2018}. More precisely, the value  $-0.28$ lies between the values obtained for the so-called weak ($\\kappa = - 0.25$) and strong chaos ($\\kappa = - 0.3$) regimes of these $1$D models having a single DoF per lattice site. \n\n\\section{Summary and conclusions} \\label{sec4} \n\nWe have studied numerically energy spreading and chaos in a nonlinear disordered architected mechanical lattice. The lattice under consideration describes rotating LEGO bricks connected with flexible links which was recently studied experimentally. The in-plane motions of the lattice are described by two DoFs per lattice site i.e., translations and angular deflections. Furthermore, in the linear limit of the aligned structure, the two DoFs per site are completely decoupled. In this state, the lattice shows two distinct behaviors corresponding to the FPUT and KG-like behaviors for translational and rotational DoFs respectively. For both cases, we review results regarding the behavior of energy spreading  under the effect of disorder.\n\nUsing single site angular deflections and time derivatives of angular deflections initial excitations, we studied the system for different strengths of nonlinearity focusing on the weakly nonlinear and strongly nonlinear regimes. For the weakly nonlinear regime we have shown that the total energy density of the lattice is split into two parts: i) a slow spreading part around the excitation point following the KG-like behavior and ii) the fast propagating tails of lower energy which travel with the speed of sound of the corresponding FPUT lattice and are responsible for the evolution of the second moment of the total energy distribution. \nFor sufficiently large initial excitations, the strong nonlinearity of the flexible architected lattice forces the initial wave-packet to spread ballistically and the distinction between a KG- and an FPUT-like behavior is lost. We note here that a ballistic behavior under strong disorder is not easily achieved and here the responsible  physical mechanism is the large geometrical nonlinearity. \n\nAdditionaly, we show that chaos is found to persist during the energy spreading although its strength decreases in time as quantified by the evolution of the system's ftMLE. Here, the power law time evolution of the exponent of the ftMLE is found to acquire a value which lies between the ones obtained for the so-called weak and strong chaos regimes of the well studied nonlinear KG lattice.\n\nOur results show that flexible architected elastic lattices with coupled DoFs per site provide a modern physical platform to study and observe rich wave dynamics which can not otherwise be seen with classical uncoupled fundamental models. Some interesting directions arise from our results like the study and manipulation of energy propagation by tuning the dispersion characteristics of rotations, by changing the shear and bending stiffness's. Furthermore, here we only considered an aligned structure where the two DoFs per site are uncoupled in the linear limit. Extensions to other geometries, where the linear modes are polarised will probably reveal a variety of spreading characteristics and provide a means of controlling energy transport in highly heterogeneous lattices. \n\n\\section*{Acknowledgements}\nA.N. acknowledges funding from the University of Cape Town (University Research Council, URC) postdoctoral Fellowship\ngrant and the Oppenheimer Memorial Trust (OMT) postdoctoral Fellowship grant. Ch. S. thanks the Universit\\'{e} du Mans for its hospitality during his visits when part of this work was carried\nout. We also thank the Centre for High Performance Computing \\cite{chpc} for providing\ncomputational resources for performing significant parts of this paper's computations. We also thank V. Tournat for useful discussions.\n\n\\section*{Author Declarations}\nThe authors have no conflicts to disclose.\n\n\\section*{Data Availability Statement}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec_intro}\n\n\n\nIn four-dimensional General Relativity, a fundamental connection between geometric optics and the algebraic structure of the Weyl tensor is provided by the Goldberg-Sachs (GS) theorem \\cite{GolSac62,NP}, which states that an Einstein spacetime is algebraically special if, and only if, it contains a shearfree geodetic null congruence (cf.~\\cite{Stephanibook,penrosebook2} for related results and generalizations). This theorem plays an important role in the construction of algebraically special exact solutions of the Einstein equations \\cite{Stephanibook}, as the remarkable discovery of the Kerr metric shows \\cite{Kerr63}. \n\nIn recent years, the growing interest in higher dimensional gravity has motivated the study of extensions of the above concepts to  $n>4$ dimensions. An algebraic classification of the Weyl tensor based on the notion of Weyl Aligned Null Directions (WANDs) has been put forward in \\cite{Coleyetal04} (see also \\cite{OrtPraPra12rev} for a recent review). Furthermore, a higher dimensional version of Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms have been presented in \\cite{Pravdaetal04,Coleyetal04vsi,OrtPraPra07} and \\cite{Durkeeetal10}, respectively. However, simple examples reveal that neither the ``geodesic part'' nor the ``shearfree part'' of the GS theorem extend in an obvious way to higher dimensions (see \\cite{Ortaggioetal12} for a brief summary, along with the original references \\cite{MyePer86,FroSto03,Pravdaetal04,OrtPraPra07,PraPraOrt07,GodRea09,Durkee09}).\n\nThe proper formulation of the geodesic part of the higher dimensional GS theorem has been proven in \\cite{DurRea09} (see also \\cite{PraPraOrt07,Durkee09}): in particular, {\\em an Einstein spacetime admits a multiple WAND if, and only if, it admits a geodesic multiple WAND} -- hence one can restrict to geodesic multiple WANDs {(mWANDs)} with no loss of generality. Very recent work \\cite{Ortaggioetal12} has analyzed the shearfree part in five dimensions by proving necessary conditions on the form of the optical matrix $\\rhob$ (defined below in \\eqref{expmatrix})  following from the existence of an mWAND.\\footnote{See \\cite{Taghavi-Chabert11} for a different formulation of the  ``shearfree'' condition and \\cite{Ortaggioetal12} for a comparison between the two approaches.}  Contrary to the 4d case, the conditions obtained in \\cite{Ortaggioetal12} are, in general, not sufficient. In fact, it seems that in higher dimensions  conditions that are both necessary and sufficient in general do not exist.\n\nIn the present contribution we focus on $n$-dimensional Einstein (including Ricci-flat) spacetimes of type~II (type D being understood as a special subcase thereof) in more than five dimensions and work out the corresponding necessary conditions on $\\rhob$ { {\\em under the assumption that the mWAND is twistfree} (i.e., $\\rhob$ is symmetric, {equivalent to $A_{ij}=0$, cf.~\\eqref{S_A} below)}}.\\footnote{This automatically guarantees that $\\bl$ is geodesic. Except for a few remarks, we assume {that $\\rhob$ is non-zero}, since we are interested in studying its possible non-trivial forms.} \nSince the types N and III were already studied in \\cite{Pravdaetal04}, we will simply connect the results of \\cite{Pravdaetal04} to our analysis when appropriate. As it turns out, in $n>5$ dimensions the constraints on $\\rhob$ are in general not as strong as those for $n=5$ \\cite{Ortaggioetal12}.  Nevertheless, combining our results for a type II spacetime with the results for type III and type N \\cite{Pravdaetal04}  gives \n\n\n\n\\begin{theorem}[Eigenvalue structure of $\\rhob$ for $n\\ge 6$ and $A_{ij}=0$]\n\\label{prop_GSHD}\n\nIn an algebraically special Einstein spacetime of dimension $n\\ge 6$ that is not conformally flat, the $($symmetric$)$ optical matrix {$\\rhob$} of a non-twisting multiple WAND has at least {\\em one double eigenvalue}. In the following special cases stronger conditions hold and the most general permitted eigenvalue structures  are\n\\begin{enumerate}[(i)]\n\t\\item if $\\Phi^A_{ij} \\ne 0$: $\\{a,a,0,\\ldots , 0\\}$ {\\rm\\cite{Ortaggioetal12}}, \n \t\t\\label{PhiA}\n\t\\item if $\\det \\rhob\\not= 0$, $\\Phi_{ij}\\not=0$: $\\{a,a,\\ldots , a\\}$ (Robinson-Trautman, $\\Phi_{ij}\\propto \\delta_{ij}$, type D(bd)), \n\\label{RT}\n \\item if  $\\Phi_{ij}=0$ (type II(abd)): $\\{a,a,b,b,c_1,\\ldots , c_{n-6}\\}$,   \t\t\t\t\t\\label{Phi0}\n\\item\nif the type is N or III: $\\{a,a,0,\\ldots , 0\\}$ {\\rm\\cite{Pravdaetal04}}. \n \\label{NIII}\n \\end{enumerate}\n\\end{theorem}\n\n(The Weyl components $\\Phi^A_{ij}$ and $\\Phi_{ij}$ are defined below in \\eqref{bw0}.) The proof of (\\ref{RT}) and  (\\ref{Phi0}) is a new result and will be given in sections~\\ref{sec_nondeg} and  \\ref{sec_nontwist}, respectively.\nNote that in the above cases (\\ref{PhiA})--(\\ref{NIII}) the matrix $\\rhob$  possesses at least {\\em two} double eigenvalues. In particular, in six dimensions all possible cases can be listed explicitly:\n\n\\begin{corol}[Eigenvalue structure of $\\rhob$ for $n=6$ and $A_{ij}=0$]\n\\label{corol6d}\nIn a 6d algebraically special Einstein spacetime that is not confomally flat, the $($symmetric$)$ optical matrix $\\rhob$ of a non-twisting multiple WAND can have only one of the following eigenvalue structures (where $a,b $ might coincide, or vanish): (i) $\\{a,a,b,b \\}$; (ii) $\\{a,a,b,0 \\}$; (iii) $\\{a,b,0,0 \\}$. If the spacetime is type III or N then the structure is $\\{a,a,0,0\\}$.\n\n\n\n\\end{corol}\n\nNote that we do not claim that all classes compatible with theorem~\\ref{prop_GSHD} or corollary  \\ref{corol6d} are non-empty. \n\nThe structure of the paper is the following. In section~\\ref{sec_genII} we briefly summarize  results for general (possibly twisting) type II spacetimes in arbitrary dimension (already contained in \\cite{Ortaggioetal12}), in particular the so called ``optical constraint''. These will be useful  in  the rest of the paper, where, however, we limit ourselves to the non-twisting case. \nIn section~\\ref{sec_nontw_gen} we set up the study of general non-twisting type II spacetimes. It turns out that it is convenient to study the case of non-degenerate and degenerate $\\rhob$ separately. This is done in sections~\\ref{sec_nondeg} and \\ref{sec_nontwist}, where we derive certain necessary conditions for $\\rhob$ being  compatible with a non-twisting mWAND, \nin particular proving thus theorem~\\ref{prop_GSHD}. Using various explicit examples we show in section~\\ref{sec_counter} that the necessary conditions obtained in sections~\\ref{sec_nondeg} and \\ref{sec_nontwist} are in general not  sufficient. In section~\\ref{sec_totgeod} we elucidate the geometrical implications of theorem~\\ref{prop_GSHD} in terms of existence of integrable null distributions with\ntotally geodesic integral null two-surfaces. In section~\\ref{sec_6D} we study the six-dimensional case (corollary  \\ref{corol6d}) and provide various explicit examples of such metrics.\n\nSome related results are given in the appendices. {{In appendix~\\ref{app_alg_eq} we present new algebraic constraints for general type II Einstein spacetimes. Appendix~\\ref{app_type_N_III} contains a new proof of the canonical form of the optical matrix $\\rhob$ for type III Einstein spacetimes in the case of a non-twisting mWAND $\\bl$ (the original, longer proof was given in \\cite{Pravdaetal04}). In appendix~\\ref{app_shearfree} we summarize various equivalent formulations of the ``geodesic and shearfree'' condition in four dimensions, and we discuss some extensions of those to higher dimensions (including optical structure, optical constraint, integrability of certain null distributions). Appendix~\\ref{app_OS5d} contains the proof (mostly relying on results of \\cite{Ortaggioetal12,Wylleman_priv}) of the existence of an optical structure in all algebraically special spacetimes in five dimensions. In appendix~\\ref{app_shearfreetwist} we present examples of twisting but shearfree mWANDs (in six dimensions). Finally, in appendix~\\ref{app_violating}  examples of Einstein spacetimes of type D with non-degenerate mWANDs violating the optical constraint are given.}}\n\n\n\n\n\\paragraph{Notation}\n\n\n\nThroughout the paper we use higher dimensional GHP formalism developed in \\cite{Durkeeetal10}. We employ a null frame\n\\begin{equation}\n  \\{\\lb \\equiv \\eb_{(0)}=\\eb^{(1)},\n  \\bn \\equiv \\eb_{(1)} = \\eb^{(0)},\n    \\mmb{i}\\equiv\\eb_{(i)} = \\eb^{(i)} \\}\n\\end{equation}\nwith indices $i,j,k,\\ldots$ running from $2$ to $n-1$. The null vector fields  $\\lb$ and $\\bn$  and the orthonormal spacelike vector fields $\\mmb{i}$  obey $\\lb \\cdot \\bn =1$, $\\mmb{i} \\cdot \\mmb{j} = \\delta_{ij}$ and  $\\lb \\cdot \\mmb{i} = 0 = \\bn \\cdot \\mmb{i}.$\nThe optical matrix $\\rhob$ is defined by\n\\be\n \\rho_{ij} = m_{(i)}^a m_{(j)}^b \\nabla_b \\ell_a , \\label{expmatrix}\n\\ee\nand its trace gives the expansion scalar (up to normalization)\n\\be\n \\rho \\equiv \\rho_{ii}.\n\\ee\n{Its (anti-)symmetric parts are, respectively,\n\\be\n S_{ij}\\equiv \\rho_{(ij)} , \\qquad A_{ij}\\equiv \\rho_{[ij]} .\n \\label{S_A}\n\\ee \nWe also define the rank of $\\rhob$ as} \n\\be\n m\\equiv\\mbox{rank}(\\rhob).\n\\ee\nOther Ricci rotation coefficients used in the paper are \\cite{Pravdaetal04,OrtPraPra07,Durkeeetal10} \n\\be\n L_{1i} = n^a m_{(i)}^b \\nabla_b \\ell_a , \\quad \\tau_i\\equiv L_{i1} =  m_{(i)}^a n^b \\nabla_b \\ell_a ,\\quad \\M{i}{j}{0}=m_{(j)}^a\\ell^b \\nabla_b m_{(i)a} , \\quad \\M{i}{j}{k}=m_{(j)}^am_{(k)}^b \\nabla_b m_{(i)a} .\n \\label{L1i_M}\n\\ee\n\n\nBoost weight (b.w.) zero components of the Weyl tensor  \n\\be\n\\Phi_{ijkl} = C_{ijkl}, \\  \\ \\  \\Phi_{ij} = C_{0i1j},\\ \\ \\  2\\Phia_{ij} = C_{01ij}, \\ \\ \\  \\Phi = C_{0101}\n\\label{bw0}\n\\ee\nare subject to\n\\bea\n&& \\Phi_{ijkl} = \\Phi_{[ij][kl]} = \\Phi_{klij}, \\qquad  \\Phi_{i[jkl]}=0, \\label{Phicycl} \\\\\n&& \\Phis_{ij} \\equiv \\Phi_{(ij)}   = -\\pul\\Phi_{ikjk}, \\qquad \\Phia_{ij} \\equiv \\Phi_{[ij]}, \\qquad \\Phi=\\Phi_{ii}, \\label{PhiS} \n\\eea\nand thus they are determined by the $\\frac{1}{12}(n-1)(n-2)^2(n-3)$ components of $\\Phi_{ijkl}$ and the $\\pul (n-2)(n-3)$ components of $\\Phia_{ij}$. In a type II spacetime, at least one of the latter two quantities must be non-vanishing when $\\bl$ is an mWAND (otherwise the spacetime would be type III or more special). \nRecall that the algebraic subtypes II(a), II(b), II(c) and II(d) are defined by the vanishing of $\\Phi$, the symmetric traceless part of $\\Phi_{ij}$, the ``Weyl part'' of $\\Phi_{ijkl}$, and $\\Phia_{ij}$, respectively (see \\cite{OrtPraPra12rev}). \n\n\nThere is no summation in case one or both repeated indices is\/are  in round brackets, unless it is said otherwise. E.g., there is no summation in $\\Phi_{i(j)k(j)}$ and there is summation in $\\Phi_{ijkj}$ and $\\sum_j \\Phi_{i(j)k(j)}$. \n\n\n{In what follows we shall employ the Sachs equation \\cite{Pravdaetal04,OrtPraPra07,Durkeeetal10}\n\\begin{eqnarray}\n  \\tho \\rho_{ij} &=& - \\rho_{ik} \\rho_{kj} \\label{Sachs},\n\\end{eqnarray}\nand the following  Bianchi identities (eqs.~(A10), (A11) of \\cite{Durkeeetal10})    \n \\begin{eqnarray}\n  \\tho \\Phi_{ij}   &=& -(\\Phi_{ik} + 2\\Phia_{ik} + \\Phi \\delta_{ik}) \\rho_{kj}, \\label{A2}\\label{Bi2}\\\\[3mm]\n  -\\tho \\Phi_{ijkl}   &=& 4\\Phia_{ij} \\rho_{[kl]} - 2\\Phi_{[k|i} \\rho_{j|l]} + 2\\Phi_{[k|j} \\rho_{i|l]} \n                       + 2\\Phi_{ij[k|m} \\rho_{m|l]} . \\label{A4}\\label{Bi3}\n\\end{eqnarray}\nIn a parallelly propagated frame,  the GHP directional derivative  $\\tho$ reduces to the NP derivative along the $\\bl$ direction, $D=\\ell^a\\nabla_a$. \nFor an affinely parametrized $\\bl$ with an affine parameter $r$ this is simply $D=\\partial_r$.}\n\n\n\n\n\n\n\nThere are two possible approaches to studying consequences of the Bianchi and Ricci equations. The first approach consists in applying the derivative operator $\\tho$ on certain algebraic equations and in deriving new algebraic constraints, e.g. eq.~\\eqref{thornB8}, using \\eqref{Sachs}--\\eqref{Bi3} (cf. also appendix~\\ref{app_alg_eq} and \\cite{Ortaggioetal12} (section~3.1 and appendix A therein)). In this paper we mainly use the second approach, which consists in solving the Sachs and Bianchi differential equations \\eqref{Sachs}--\\eqref{Bi3} and then in analyzing the compatibility of their solutions with the algebraic equations \\eqref{B4}--\\eqref{thornB8}.\n\n\n\\section{Results for general type II}\n\\label{sec_genII}\n\n\nThis section is devoted to summarizing some useful results that hold for all type II Einstein spacetimes, without assuming that $\\rhob$ is non-twisting. In particular we present the algebraic restrictions on the Weyl components of b.w. zero that follow from the Bianchi identities, to be used in the next sections. More general new results are given in appendix~\\ref{app_alg_eq} for future reference.\nFor genuine type II spacetimes the (unique) mWAND is necessarily geodesic, while for type D spacetimes there always exists a geodesic mWAND \\cite{DurRea09}. In the frame used below we thus take $\\bl$ to be a geodesic mWAND, without loss of generality. \n \n\n\\subsection{Algebraic constraints}\n\n\nFor type II Einstein spacetimes, the Weyl tensor must obey the algebraic constraints (A.12) and\n(B8) of \\cite{Durkeeetal10} (already discussed in \\cite{PraPraOrt07,Durkee09}) and $\\tho$(B8) (derived in \\cite{Ortaggioetal12}), namely \n\\bea\n2\\Phia_{[jk|}\\rho_{i|l]} -2\\Phi_{i[j}\\rho_{kl]} + \\Phi_{im [jk|}\\rho_{m|l]}  = 0, \\label{B4} \\\\\n\\Phi_{kj} \\rho_{ij} - \\Phi_{jk} \\rho_{ij} + \\Phi_{ij} \\rho_{kj} - \\Phi_{ji} \\rho_{jk} \n       + 2 \\Phi_{ij} \\rho_{jk} - \\Phi_{ik} \\rho  + \\Phi \\rho_{ik} + \\Phi_{ijkl} \\rho_{jl} = 0 \\label{B8} , \\\\\n\\left( 2 \\Phi_{kj} - \\Phi_{jk} \\right) \\rho_{il} \\rho_{jl} + \\left( 2 \\Phi_{ij} - \\Phi_{ji} \\right) \\rho_{jl} \\rho_{kl}  - \\Phis_{ik} \\rho_{jl} \\rho_{jl} + \\Phi \\rho_{il} \\rho_{kl} + \\Phi_{ijkl} \\rho_{js} \\rho_{ls} =0. \\label{thornB8}\n\\eea\n\n\nEq. \\eqref{B8} is traceless and its\nsymmetric and antisymmetric parts read, respectively,\n\\bea\n\\left( 2 \\Phi_{kj} - \\Phi_{jk} \\right) S_{ij} + \\left( 2 \\Phi_{ij} - \\Phi_{ji} \\right) S_{jk}  - \\Phis_{ik} \\rho + \\Phi S_{ik} + \\Phi_{ijkl} S_{jl} =0, \\label{B8sym}\\\\\n\\Phi_{jk} A_{ji} + \\Phi_{ji} A_{kj} + \\Phi_{ij} \\rho_{jk} -\\Phi_{kj} \\rho_{ji} +  \\Phia_{k i} \\rho + \\Phi A_{ik} + \\Phi_{ijkl} A_{jl} = 0 .   \\label{B8asym} \n\\eea\n\n\n\nThe reason for investigating algebraic conditions such as \\eqref{B4}--\\eqref{B8asym} (or those of \\cite{Pravdaetal04} in the case of type III\/N) is that they will in general constrain the possible form of $\\rhob$. This way one obtains the standard Goldberg-Sachs theorem in four dimensions and one can arrive at similar conclusions also in higher dimensions, at least with additional assumptions (e.g. on the Weyl type \\cite{Pravdaetal04}, on the number of dimensions \\cite{Ortaggioetal12}, \non the form of the line-element \\cite{OrtPraPra09,MalPra11}, on the asymptotic behaviour \\cite{OrtPraPra09b}, and\/or on optical properties of $\\bl$, as we shall discuss in the next sections).\n\n\n\\subsection{The optical constraint}\n\n\\label{subsec_OC}\n\n\nIt was observed in \\cite{Ortaggioetal12} that the above conditions on the Weyl tensor appear to be less stringent when $\\rhob$ satisfies\nthe {\\em optical constraint} \\cite{OrtPraPra09}\n\\be\n\\rho_{ik} \\rho_{jk} \\propto \\rho_{(ij)} . \\label{OC2}\n\\ee\nIn particular, when this holds eq.~\\eqref{thornB8} is not an extra restriction. \n{This thus suggests that the branch of type II solutions {whose mWAND} obeys \\eqref{OC2} corresponds to the case of a ``generic'' Weyl tensor \\cite{Ortaggioetal12}.}\nIt has been proven that \\eqref{OC2} indeed holds for {the Kerr-Schild vector} of all (generalized) Kerr-Schild spacetimes \\cite{OrtPraPra09,MalPra11} (including Myers-Perry black holes \\cite{MyePer86}), for non-degenerate geodesic double WANDs in asymptotically flat type II spacetimes \\cite{OrtPraPra09b} (see eq.~(14) therein) {and for the unique double WAND of all genuine type II Einstein spacetimes in five dimensions (see footnote~\\ref{foot_typeD} for the type D case) \\cite{Ortaggioetal12,Wylleman_priv}.} \nNevertheless, {double WANDs} violating the optical constraint also exist, as shown by explicit examples constructed below in section~\\ref{subsubsec_examples_distinct} for $n\\ge 6$, in appendix~\\ref{app_violating} for $n\\ge 7$, and in section~6.3 of \\cite{Ortaggioetal12} for $n=5$.\\footnote{We observe that all such examples are of type D. In fact, in five dimensions {\\em all} type D Einstein spacetimes admitting a geodesic mWAND violating the optical constraint are known \\cite{Ortaggioetal12} (and coincide with the class of Einstein spacetimes admitting a non-geodesic mWAND \\cite{DurRea09}). Such ``exceptional'' null directions are necessarily twisting in 5d (but not in higher dimensions). \nHowever, in all such 5d type D spacetimes there always also exists a pair of non-aligned (non-twisting) mWANDs that {\\em do} obey the optical constraint \\cite{Ortaggioetal12}. On the other hand, this is generically not true when $n>5$ (see appendix \\ref{subsubsec_ex2} for a ten-dimensional example admitting exactly two mWANDs, both violating the optical constraint).\\label{foot_typeD}}\nAlthough the above analysis applies only to type D and genuine type II  spacetimes (eqs.~(\\ref{B4})--(\\ref{thornB8}) become trivial identities for more special types), it is worth remarking that the optical constraint is also obeyed by the mWAND of all type N Einstein spacetimes \\cite{Pravdaetal04,Durkeeetal10}. {The mWAND $\\bl$ of type III Einstein spacetimes also obeys the optical constraint provided either \\cite{Pravdaetal04}: (i) the spacetime is five-dimensional; (ii) the Weyl tensor satisfies a genericity condition (see~\\cite{Pravdaetal04}); (iii) $\\bl$ is  {\\it non-twisting} (see also appendix \\ref{app_type_N_III} for a simpler proof in case (iii))}. \n\n \nThe optical constraint implies that $(\\Id- \\frac{2}{\\alpha} \\rhob)$ is an orthogonal matrix \\cite{Ortaggioetal12} and that consequently  $[\\rhob, \\rhob^T]$ vanishes \\cite{OrtPraPra09,Ortaggioetal12}. The optical matrix $\\rhob$ is therefore a {\\em normal} matrix and can thus be put, using spins,  into a convenient block-diagonal form {(see \\cite{OrtPraPra09,OrtPraPra10} for extended related discussions)}, i.e.,\n\\bea\n\\rhob = \\alpha{\\rm diag}\\left(1, \\dots 1, \n\\frac{1}{1+ \\alpha^2 b_1^2}\\left[\\begin {array}{cc} 1 & -\\alpha b_1 \\\\ \n   \\alpha b_1 & 1  \\label{canformL} \\\\\n  \\end {array}\n \\right]\n, \\dots, \n\\frac{1}{1+ \\alpha^2 b_\\nu^2} \\left[\\begin {array}{cc} 1 & -\\alpha b_\\nu \\\\ \n   \\alpha b_\\nu & 1 \\\\ \\end {array}\n \\right] \n , 0, \\dots ,0\n\\right).\n\\eea\nThe block-diagonal form \\eqref{canformL} {is useful for practical purposes because it} allows for determining the $r$-dependence of $\\rhob$ by integrating the Sachs equation \\eqref{Sachs} {(in a parallelly transported frame)} \\cite{OrtPraPra10}. {\\em In the non-twisting case it reduces to a sequence of 1s followed by a sequence of 0s} (up to an overall factor). Note that the symmetric part of each two-block is proportional to a two-dimensional identity matrix, i.e. it is ``shear-free''. In four dimensions the optical constraint implies that either there is a single such block (in which case $\\rho_{ij}$ is shearfree) or that $\\rho_{ij}$ is symmetric with exactly one non-vanishing eigenvalue. However, the Goldberg-Sachs theorem shows that the latter case cannot occur. Therefore in 4d the optical constraint is a necessary condition for $\\bl$ to be a repeated principal null direction but it is not sufficient \\cite{Ortaggioetal12}. \n\n \nOn the other hand, in higher dimensions a vector field $\\bl$ obeying the optical constraint is in general {\\em shearing}, as in the case of Myers-Perry black holes \\cite{MyePer86} (see also \\cite{FroSto03,PraPraOrt07} for a discussion of their optical properties). Together with other results \\cite{Pravdaetal04,PodOrt06,OrtPraPra07}, this has made clear that the shearfree condition is in general ``too restrictive'' in higher dimensions, as opposed to the four-dimensional case. In particular, it was observed in \\cite{OrtPraPra07} that in {\\em odd} dimensions a twisting geodesic mWAND is necessarily shearing. By contrast, twisting geodesic mWANDs with zero shear are permitted in {\\em even} dimensions and they have necessarily $\\det(\\rhob)\\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\\propto\\delta_{ij}$). An explicit example in six dimensions is discussed in appendix \\ref{app_shearfreetwist}.\n\n\n\nIn the non-twisting case, the existence of shearfree spacetimes has been already known for some time in all dimensions -- they are either \\RT \\cite{PodOrt06} or Kundt \\cite{PodZof09} solutions, according to the presence\/absence of expansion.\n \n\n\n\n\n\n\\subsection{Possible generalizations of the geodesic{\\&}shearfree property}\n\n\nIn arbitrary dimensions various geometric conditions can be considered which are different from the standard geodesic{\\&}shearfree condition (considered above) and which, however, become all equivalent (except for the optical constraint, cf.~\\cite{Ortaggioetal12}) in the special case $n=4$.  Further such conditions are discussed in appendix~\\ref{app_shearfree}. \nOne could thus conceive that various formulations of the Goldberg-Sachs theorem are in principle possible when $n>4$. Some of these formulations have been studied in \\cite{Taghavi-Chabert11,Taghavi-Chabert11b,Ortaggioetal12} (see also \\cite{OrtPraPra09}) but none of these gives necessary and sufficient conditions. In the rest of this paper we will discuss necessary conditions determined by the presence of a non-twisting mWAND in an $n$-dimensional Einstein spacetime and we will present several explicit examples. A possible interpretation of these results in terms of the geometric conditions of appendix~\\ref{app_shearfree} will be discussed in sections~\\ref{sec_totgeod} (for $n\\ge6$) and \\ref{subsec_integrability} (for $n=6$).\n\n\n\n\n\n\\section{Non-twisting $\\bl$: general properties}\n\\label{sec_nontw_gen}\n\n\n\nHere we study the optics of a hypersurface orthogonal mWAND $\\bl$ in a type II Einstein spacetime.  This mWAND is automatically geodetic. Thus we have \n\\be\n \\kappa_i=0=A_{ij} ,\n\\ee\n{so that $\\rho_{ij}=S_{ij}$.} Since the algebraic equations \\eqref{B4}--\\eqref{B8asym}  are trivial for  Kundt spacetimes, in what follows we assume $\\rhob\\not= ${\\bf 0}.\n\n\n\n\\subsection{Case $\\Phia_{ij} \\neq 0$}\n\n\\label{sec_PhiA}\n\nThis case has been already analyzed in \\cite{Ortaggioetal12}, {arriving at} point (i) of theorem~\\ref{prop_GSHD} of section~\\ref{sec_intro}. As already observed there, in this case $\\rhob$ satisfies the optical constraint and it is  necessarily degenerate ($m=2$).  Moreover it is shearing, except for $\\rhob=0$ {(which is necessarily the case for $n=4$)}.\n\nIt remains to consider the case when $\\Phi^A_{ij}=0$. \n\n\n\\subsection{Case $\\Phia_{ij}=0$}\n\n\nThis case defines the subtype II(d) {in the notation of \\cite{Coleyetal04}}. For  $\\Phia_{ij}=0$, eq.~\\eqref{B8} reduces to  (recall $A_{ij}=0$ here)\n\\be\n -\\rho \\Phi_{ik} + \\Phi \\rho_{ik} + 2\\Phi_{ij} \\rho_{jk} + \\Phi_{ijkl} \\rho_{jl}=0 ,  \\label{eqn:algI}\n\\ee\nand eq. \\eqref{B8asym}  to \n\\be\n[\\rhob,\\Phib^S]=0.\\label{comm_rho_Phi}\n\\ee\nNote that taking into account  (\\ref{comm_rho_Phi}), eq. (\\ref{eqn:algI}) is symmetric and corresponds to eq.~\\eqref{B8sym}.\n\nSimilarly, eq. \\eqref{thornB8} yields \n\\be\n -\\Phi_{ik} \\rho_{jl} \\rho_{jl} + \\Phi (\\rho^2)_{ik} + 2 \\Phi_{ij} (\\rho^2)_{jk} + \\Phi_{ijkl} (\\rho^2)_{jl} = 0\\label{eqn:rhoC}.\n \\ee\n\n  \n\nThanks to \\eqref{comm_rho_Phi}, one can choose a {basis where both $\\rho_{ij}$ and $\\Phi_{ij}$ are diagonal}, $\\rho_{ij}=\\diag(\\rho_2,\\rho_3,\\dots)$,  \n$\\Phi_{ij}=\\diag(\\Phi_2,\\Phi_3,\\dots)$, { therefore $\\Phi_{ijkj}=0$ for $i\\neq k$}.\nIn this frame, {the off-diagonal components of the algebraic constraint~(\\ref{eqn:algI}) are} \n\\be \n \\sum_j\\rho_{(j)}\\Phi_{i(j)k(j)}=0 \\qquad (\\mbox{for } k\\neq i) .\n \\label{nondiag_constr}\n\\ee\n The diagonal part of~(\\ref{eqn:algI}) can be expressed as  ${\\cal L}_{ij} \\tilde \\rho_j =0$, \nwhere $\\tilde \\rho_{i}$ is the  vector  $\\tilde \\rho_{i}=(\\rho_2, \\rho_3 \\dots)$  \nand the linear operator ${\\cal L}_{ij}$ is given by\n\\[\n  {\\cal L}_{ij} = W_{ij} + \\Phi  \\delta_{ij} + 2 \\diag (\\Phi_{2},\\Phi_{3},\\dots \\Phi_{n}) - \\left[ \n                     \\begin{array}{cccc}\n                       \\Phi_{2}                  & \\Phi_{2}        & \\dots & \\Phi_{2}     \\\\\n                       \\Phi_{3}       & \\Phi_{3}       & \\dots & \\Phi_{3}   \\\\\n                       \\vdots    & \\vdots    & \\dots & \\vdots\\\\\n                       \\Phi_{n} & \\Phi_{n} & \\dots & \\Phi_{n}\n                     \\end{array}\n               \\right]\n\\]\nor, equivalently, ${\\cal L}_{ij} = W_{ij} +( \\Phi + 2\\Phi_{(i)} )  \\delta_{ij} -\\Phi_{(i)} $, where {we have defined the symmetric and traceless matrix}\n\\be\n\tW_{ij} \\equiv C_{(i)(j)(i)(j)} \\qquad {\\mathrm{\\ \\ (no\\ summation)}}.\\label{def_W}\n\\ee\n{Note that $W_{(i)(i)}=0$.} From  the first of \\eqref{PhiS}, it follows  \n\\be\n\\sum_j W_{ij} = - 2 \\Phi_i ,\n \\label{W_phi}\n\\ee\n{which will be used in certain calculations throughout the paper.} (From  \\eqref{W_phi} \nit follows that $W_{ij}=0\\Rightarrow {\\cal L}_{ij}=0$.)  \nThus the sum of the rows of ${\\cal L}_{ij}$ vanishes and so they are  linearly dependent and \n\\be\n\\det {\\cal {\\bm{L}}} =0. \\label{detL}\n\\ee \nTherefore  zero is an eigenvalue of ${\\cal L}$ and  (the diagonal part of) (\\ref{eqn:algI})  admits non-trivial solutions.\n\nNote that (\\ref{eqn:rhoC}) has the form ${\\cal L}_{ij} {\\tilde \\rho}^2_j =0$, where components of  the $(n-2)$ dimensional vector ${\\tilde \\rho}^2$ are squares of components of ${\\tilde \\rho}$.  The characteristic polynomial of ${\\cal L}$ is \n\\be\n \\det({\\cal L}-\\lambda I)=\\lambda^{n-2} + k_{n-3} \\lambda^{n-3} + \\dots + k_1 \\lambda + k_0 ,\n\\ee\nwith $k_0=0$ due to (\\ref{detL}).\nNow zero is a multiple eigenvalue of ${\\cal L}$ iff  $k_1=0$. Let us observe that for a generic form of a type II Weyl tensor $k_1$ is non-vanishing {and thus zero is a single eigenvalue \nof ${\\cal L}$}.\\footnote{For example in five dimensions $k_1 = 12 \\left(\\phi_2 \\phi_4+ \\phi_3 \\phi_4 + \\phi_2 \\phi_3 \\right)$. {If we consider, for instance, five-dimensional} \\RT spacetimes ({which coincide with the Schwarzschild solution plus a possible cosmological constant}), characterized by $\\rho_{ij} \\propto \\delta_{ij}$,  equation \\eqref{eqn:algI} implies $\\Phi_{ij} \\propto \\delta_{ij}$, which clearly leads to $k_1 \\not=0$.} \nIf this is the case then   ${\\tilde \\rho}^2$ is proportional to ${\\tilde \\rho}$ and therefore \n\\be\n\\rho_{ij}= \\alpha \\diag(1,1, \\dots 1,0,\\dots 0).\n\\ee\nRecalling also point (i) of theorem~\\ref{prop_GSHD},\nwe can conclude with \n\n\\begin{prop}\nFor non-twisting type II Einstein spacetimes with $k_1 \\not=0$, all non-vanishing  eigenvalues of the optical matrix $\\rho_{ij}$ coincide. \n\\label{prop_nontwist}\n\\end{prop}\n\nIf $\\Phia_{ij} \\not= 0$, the stronger result of point (i) of theorem~\\ref{prop_GSHD} holds.\n\n\n\nAs we will see in sections~\\ref{subsec_alldistinct} and \\ref{subsec_permitted_6D} (table~\\ref{tab_6D}), \nthere do exist non-twisting type II Einstein spacetimes with distinct non-vanishing eigenvalues of $\\rhob$. \nAccording to Proposition~\\ref{prop_nontwist}, for these spacetimes $k_1$ vanishes. In particular $\\cal L$, and therefore also $k_1$, vanishes for spacetimes with $W_{ij}=0$. We will study special cases with various parts of the Weyl tensor vanishing in the following sections.\n\n\n\\section{Non-twisting $\\bl$: non-degenerate $\\rhob$ ($m=n-2$)}\n\n\n\\label{sec_nondeg}\n\nFirst, let us consider the case of a non-degenerate $\\rhob$ (i.e., $\\det\\rhob\\not= 0$ -- this is relevant, e.g., for asymptotically flat algebraically special spacetimes \\cite{OrtPraPra09b}). When $\\det\\rhob\\not= 0$ one necessarily has (see point (i) of theorem~\\ref{prop_GSHD}) \n\\be\n \\Phia_{ij}=0 ,\n\\ee\nso that the type is II(d). As noticed above we can use a frame in which both $\\rho_{ij}$ and $\\Phi_{ij}$ are diagonal, which is moreover compatible with parallel transport \\cite{PraPra08,OrtPraPra10}. Thus the eigenvalues of the optical matrix  are \\cite{PraPra08}\n\\be\n \\rho_i=\\frac{1}{r-b_i} .\n \\label{rho_i}\n\\ee\n\n{In order to discuss the possible structures of $\\rhob$ it is convenient to discuss separately various cases in which different parts of the Weyl tensor are (non-)zero.}\n\n\n\\subsection{Cases $\\Phi_{ij}\\neq0$ and $\\Phi_{ij}=0\\neq W_{ij}$}\n\n\\label{subsec_nondeg_Phi}\n\nFrom Bianchi equation~\\eqref{Bi2}  one has\n\\beqn\n D\\left(\\frac{\\Phi_{(i)}}{\\rho_{(i)}}\\right)=-\\Phi .\n \\label{nondeg_DPhi_i}\n\\eeqn\nThe case when all  $\\rho_i$ coincide (and are non-zero) is the known \\RT case, for which also all  $\\Phi_i$ must coincide {\\cite{PraPra08,PodOrt06}.}\n{Thus} let us  consider here the case when at least two eigenvalues $\\rho_i$ are different. \n\n\nSince the r.h.s. {of~(\\ref{nondeg_DPhi_i})} is the same for any $i$, we obtain that either all $\\Phi_i=0$, or\n\\be\n \\Phi_{i}=\\rho_i(A+\\Phi^0_{(i)}) ,\n \\label{Phi_i}\n\\ee\nwhere $A=A(r)$ is a function that must satisfy the equation $DA+A\\rho=-\\Phi^0_{j}\\rho_j$ (since $\\Phi=A\\rho+\\Phi^0_{j}\\rho_j$) and, from now on, quantities with a superscript $^0$ do not depend on $r$. Using the intermediate substitution $A(r)=Y(r)\\prod_k\\rho_k$, one arrives at the solution \n\\be\n A=\\left(\\prod_k\\frac{1}{r-b_k}\\right)\\left[-\\sum_j\\Phi^0_{j}\\int\\d r\\prod_{l\\neq j}(r-b_l)+A^0\\right] .\\label{A_nondeg}\n\\ee\nFor subsequent discussions it is useful to rewrite \\eqref{A_nondeg} using partial fraction decomposition, i.e., \n\\be\nA=\\frac{P_{n-2}}{\\prod_k {(r-b_k)}}=a^0+\\frac{P_{n-3}}{\\prod_k {(r-b_k)}}=a^0+\n\\sum_{K=1}^{\\alpha_{max}} \\sum_{\\forall i,  \\alpha_i\\geq K}\\frac{c^{(K)}_{(i)}}{(r-b_i)^K} ,\\label{A_decomposed}\n\\ee\nwhere $P_k$ denotes a polynomial of order $k$ in $r$,  $c^{(K)}_{(i)}$ {do not depend on $r$} and $\\alpha_i$ denotes the multiplicity of $b_i$, with $\\alpha_{max}$ being the maximal multiplicity. {In particular, the term $a^0$ can be determined by looking at} the leading term of \\eqref{A_nondeg} in the limit $r\\to\\infty$, i.e., \n\\be\n A=-\\frac{1}{n-2}\\sum_k\\Phi^0_k+O(r^{-1}) ,\n\\ee\nso that\n\\be\na^0=-\\frac{1}{n-2}\\sum_k\\Phi^0_k.\\label{am}\n\\ee\nIn the case $\\Phi^0_j=0$, $a^0$ vanishes and  one has  simply $A=A^0\/\\prod_k {(r-b_k)}$. \n\n\nNext, the equation for $W_{ij}$ (see \\eqref{Bi3}) can be written as (but recall $W_{(i)(i)}=0$)\n\\beqn\n & & D\\left(\\frac{W_{ij}}{\\rho_{(i)}-\\rho_{(j)}}\\right)=\\frac{\\Phi_{i}\\rho_j+\\Phi_{j}\\rho_i}{\\rho_{(i)}-\\rho_{(j)}} \\qquad (\\rho_{j}\\neq\\rho_{i}) , \\\\\n & & D\\left(\\frac{W_{ij}}{\\rho_{(i)}^2}\\right)=\\frac{\\Phi_{i}+\\Phi_{j}}{\\rho_{(i)}} \\qquad (\\rho_{j}=\\rho_{i}) .\n\\eeqn\nUsing the above expression for $\\Phi_i$, in both cases one can write the solution as\n\\be\n W_{ij}=\\frac{1}{(r-b_{(i)})(r-b_{(j)})}\\left[2\\int\\d r A+(\\Phi^0_{i}+\\Phi^0_{j})r+ W^0_{ij}\\right] .\\label{Wij_nondeg}\n\\ee\n\nNow, imposing $-2\\Phi_i=\\sum_jW_{ij}$ {(recall~(\\ref{W_phi}))} we obtain the constraint\n\\be\n -2(A+\\Phi^0_{i})=\\sum_{j\\neq i}\\frac{1}{r-b_{(j)}}\\left[2\\int\\d r A+(\\Phi^0_{i}+\\Phi^0_{j})r+ W^0_{ij}\\right] .\n \\label{constrW}\n\\ee\n{For $r\\to\\infty$, using \\eqref{A_decomposed} and \\eqref{am} at \nthe leading order (\\ref{constrW}) implies that all  $\\Phi^0_{i}$ coincide, i.e., }\n\\be\n \\Phi^0_{i}=\\frac{1}{n-2}\\sum_k\\Phi^0_k\\equiv{f^0}. \n \\label{Phi_i0}\n\\ee\n$A$ thus becomes\n\\be\n A=-f^0+A^0\\prod_k\\frac{1}{r-b_k} ,\n \\label{A_2}\n\\ee\nwhere we have used $\\sum_j\\prod_{l\\neq j}(r-b_l)=D[\\prod_k(r-b_k)]$, so that\n\\be\n \\Phi_{i}=\\frac{A^0}{r-b_{i}}\\prod_k\\frac{1}{r-b_k} , \\qquad W_{ij}=\\frac{1}{(r-b_{(i)})(r-b_{(j)})}\\left[2A^0r\\prod_k\\frac{1}{r-b_k}+ W^0_{ij}\\right] .\n \\label{nondeg_Weyl_simplif}\n\\ee\n\n\nThe constraint~(\\ref{constrW}) can thus be written as\n\\be\n -2A^0\\prod_k\\frac{1}{r-b_k}=\\sum_{j\\neq i}\\frac{1}{r-b_{(j)}}\\left[2A^0\\int\\d r\\prod_k\\frac{1}{r-b_k}+W^0_{ij}\\right] \\qquad (i=2,\\ldots,n-1) .\n \\label{constrW2}\n\\ee\n\n{It is now useful to discuss separately various subcases with different multiplicity of eigenvalues.} \n\n\\subsubsection{All the $b_i$ coincide: shearfree congruences (Robinson-Trautman spacetimes)}\n\nIn the \\RT case all the $b_i$ {of (\\ref{rho_i})} coincide {(and can be set to zero by shifting $r$, if desired)} and {from~(\\ref{constrW2})} we simply obtain $\\sum_{j}W^0_{ij}=0$. \n\nOn the other hand, let us assume in the following that not all $b_i$ coincide, i.e., at least two of these are distinct, say $b_2\\neq b_3$. \n\n\\subsubsection{Shearing case with all $b_i$ distinct: not permitted}\n\nFirst, if all  $b_i$ are distinct, {i.e., $\\alpha_{max}=1$ in \\eqref{A_decomposed}}, we can compute explicitly the required integral using partial fraction decomposition \n\\be\n \\int\\d r\\prod_k\\frac{1}{r-b_k}=\\sum_k\\frac{\\ln(r-b_k)}{\\prod_{l\\neq k}(b_k-b_l)} \\qquad \\mbox{($b_k$ all distinct)}.\n\\ee\nIt is thus clear that the singularity structure of the l.h.s. and the r.h.s. of (\\ref{constrW2}) (for any $i$) cannot be the same unless $A^0=0$. Therefore, {from~(\\ref{constrW2})} also $\\sum_{j}(r-b_{(j)})^{-1}W^0_{ij}=0$. However, this condition  implies that  $b_i$ cannot be all distinct, so that this case is in fact not permitted (unless $W^0_{ij}=0$, so that $W_{ij}=0$ and therefore also $\\Phi_{ij}=0$, contrary to our assumptions here -- however, we will see in sections~\\ref{subsec_nondeg_Phi3} and \\ref{subsec_nondeg_Phi4}  below that the case with all distinct $b_i$ is ruled out also for $W^0_{ij}=0$). \n\n\n\\subsubsection{Shearing case with at least one $b_i$ repeated}\n\\label{subsec_onerep}\n\nLet us now consider the case when at least one $b_i$ is repeated, say $b_2$, with multiplicity $1<\\alpha_2<n-2$, and there exists $b_3\\neq b_2$ (this is not a restriction since we are excluding here the \\RT case, which corresponds to $\\alpha_2=n-2$). \n\nUsing partial fraction decomposition {similarly as in}~\\eqref{A_decomposed}, the left and right hand sides {of (\\ref{constrW2}) take} the form (for $i=3$)\n\\bea\n\\mbox{lhs}&=&\n-2A^0\\left[\\frac{p_{\\alpha_2}}{(r-b_2)^{\\alpha_2}}+(\\mbox{terms with lower order poles at } b_2 \\mbox{ and with no poles at }b_2)\\right],\\nonumber\\\\\n\\mbox{rhs}&=&\n -2A^0\\frac{p_{\\alpha_2}\\alpha_2}{(\\alpha_2-1)(r-b_2)^{\\alpha_2}} +(\\mbox{terms with lower order poles at } b_2 \\mbox{ and with no poles at }b_2) , \\nonumber\n\\eea\nrespectively, where $p_{\\alpha_2}\\not= 0$. Comparing the highest order terms in $1\/(r-b_2)$ we thus  get\n\\be\n A^0=0, \\qquad \\sum_{j}\\frac{W^0_{ij}}{r-b_{(j)}}=0 .\n\\ee\n\n\nIn particular, it follows from (\\ref{nondeg_Weyl_simplif}) that \n\\be \n \\Phi_i=0 ,\n\\ee\nso that the Weyl type is II(abd), and \\eqref{Wij_nondeg} thus gives \n\\be\n W_{ij}=\\frac{W^0_{ij}}{(r-b_i)(r-b_j)} ,\n \\label{Wij}\n\\ee\nwith $W^0_{ij}=W^0_{ji}$ and $W^0_{(i)(i)}=0$.\n\nNow, because of (\\ref{Wij}) the condition $\\sum_jW_{ij}=0$ constraints the possible multiplicities of  $b_i$ (i.e., of the eigenvalues of $\\rho_{ij}$). First, if $b_i$ are all distinct, we immediately get $W_{ij}^0=0$, {as already discussed above}. Similarly, recalling $W_{ij}^0=W_{ji}^0$ it is easy to see that for $W_{ij}^0\\neq0$ the structures  $\\{a,a,c_1,\\ldots,c_{n-4}\\}$ and  \n $\\{a,a,a,c_1,\\ldots,c_{n-5}\\}$ are also forbidden.\\footnote{To see that \n $\\{a,a,a,c_2,\\ldots,c_{n-5}\\}$ cannot occur, assume $b_2=b_3=b_4$ and all remaining eigenvalues are single. Then $\\sum_jW_{ij}=0$ with (\\ref{Wij}) gives\n$W^0_{i2}+W^0_{i3}+W^0_{i4}=0$ and $W^0_{i\\mu}=0$ ($\\mu\\neq 2,3,4$). Since $W^0_{ij}=W^0_{ji}$, this implies that the only possible non-zero components are $W_{23}$, $W_{24}$ and $W_{34}$, with the conditions $W_{23}^0+W_{24}^0=0$, $W_{32}^0+W_{34}^0=0$ and $W_{42}^0+W_{43}^0=0$. However, this system admits only the solution $W_{23}^0=W_{24}^0=W_{34}^0=0$, so that $W_{ij}=0$.\\label{foot_311}}  \nHowever, the structure  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ is compatible with such constraints (take, e.g., $b_2=b_3$, $b_4=b_5$ and $W_{34}^0=-W_{35}^0=W_{25}^0=-W_{24}^0\\neq0$, {and the remaining $W_{ij}$ equal to zero}). \n\nTo summarize, we have seen above that in type II Einstein spacetimes with a non-degenerate non-twisting mWAND $\\bl$, one has $\\Phia_{ij}=0$ and\n\\begin{enumerate}\n \\item if $\\Phi_{ij}\\neq0$, then $\\bl$ must be shearfree and the corresponding spacetimes belong to the \\RT class. {This includes, in particular, the result of \\cite{OrtPraPra09b} for asymptotically flat type II vacuum spacetimes (restricted to non-twisting case)}\\footnote{Ref.~\\cite{OrtPraPra09b} used an expansion method. Instead of the condition $\\Phi_{ij}\\neq 0$, a condition on the asymptotic fall-off behaviour of the Weyl tensor was assumed there. In the present notation, that amounts to taking $W_{ij}^0=\\Phi^{\\{3\\}}_{ijkl}=\\Phi^{\\{4\\}}_{ijkl}=0$, which requires that $\\Phi_{ij}\\neq 0$ (otherwise the type would be III). The assumptions of \\cite{OrtPraPra09b} in the non-twisting case were thus stronger than those used here (note indeed that \\RT spacetimes with $W_{ij}^0\\neq 0$ do exist \\cite{PodOrt06}). {(The symbols $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ are defined in sections~\\ref{subsec_nondeg_Phi3} and \\ref{subsec_nondeg_Phi4} below.)}} \n\\item if $\\Phi_{ij}=0$ (type II(abd)) the structure of eigenvalues of $\\rho_{ij}$ can be more generic, however for $\\Phi_{ij}=0\\neq W_{ij}$ it must have the multiplicities $\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special). \n\\end{enumerate} \n\n\nLet us now discuss the remaining cases (in which, without loss of generality, we could assume $\\Phi_{ij}=0=W_{ij}$ --  however this will actually not be used to prove the following results).\n\n\n\n\n\\subsection{Case $\\Phi^{\\{3\\}}_{ijkl}\\neq0$} \n\n\\label{subsec_nondeg_Phi3}\n\n\n{Here we assume that a component of $\\Phi_{ijkl}$ with precisely three distinct values of $i,j,k,l$ is non-vanishing,\ni.e., for some $i\\not=k$, we have $\\Phi_{i(j)k(j)} \\not=0$.} This case is possible {only for $n\\ge6$} because we need at least three possible distinct values for $i,j,\\ldots$ and because of the tracefree condition $\\Phi_{ijik}=0$ with $k\\neq j$ (however, {for $n=4,5$ necessarily $\\Phi_{ij}\\neq0$} and from the above discussion one already knows that \\RT is the only possibility).\n\nEq.~\\eqref{B4}  gives\n\\be\n \\Phi_{(i)(j)(i)(k)}(\\rho_{(j)}-\\rho_{(k)})=0 \\qquad (k\\neq j) .\\label{B4offdiag}\n\\ee\nWithout loss of generality, we can assume $\\Phi_{2324}\\neq0$, so that necessarily $b_3=b_4$. \n\nNow, the off-diagonal ($i\\not=k$) components of \\eqref{Bi3} read\n\\be\n-D \\Phi_{i(j)k(j)} = \\Phi_{i(j)k(j)} \\rho_{(j)} + \\Phi_{i(j)k(j)} \\rho_{(k)}.\n\\ee\nTaking into account $\\rho_k = (r-b_k)^{-1}$, we arrive at\n\\be\n\\Phi_{i(j)k(j)} = \\frac{\\Phi^{0}_{i(j)k(j)}}{(r-b_{(j)})(r-b_{(k)})}  \\qquad (i \\not= k). \\label{Phi_ijkj}\n\\ee\n\n\nThen,    \nthe condition $\\Phi_{ijkj}=0$, $k\\not=i$, implies that at least another component $\\Phi_{(i)3(i)4}$ must be non-zero, say $\\Phi_{5354}\\neq0$, and that $b_5=b_2$. The eigenvalue structure must therefore again be  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special).\n\n\n\\subsection{Case $\\Phi^{\\{4\\}}_{ijkl}\\neq0$} \n\n\\label{subsec_nondeg_Phi4}\n\nHere we assume that there exists a non-vanishing component of $\\Phi_{ijkl}$ with all values of $i, j, k, l$ being distinct. This case is possible only for $n\\ge6$. \n\nSimilarly as in the previous case, eq.~\\eqref{Bi3} implies\n\\be\n \\Phi_{ijkl} = \\frac{\\Phi^{0}_{ijkl}}{(r-b_{(k)})(r-b_{(l)})},  \\qquad  {\\mathrm{\\ (for \\ }} i,j,k,l {\\mathrm{\\ all \\ distinct}}). \n\\ee\nHowever, since $\\Phi_{ijkl}=\\Phi_{klij}$ we obtain $b_i=b_k$ and $b_j=b_l$ (or $b_i=b_l$ and $b_j=b_k$) and the eigenvalue structure is again \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special). \n\n\\subsection{Summary}\n\nNo further cases are possible, since $\\Phi^{\\{4\\}}_{ijkl}=\\Phi^{\\{3\\}}_{ijkl}=W_{ij}=0$ implies the type III. \nTo summarize, {in sections~\\ref{subsec_nondeg_Phi}--\\ref{subsec_nondeg_Phi4}} we have thus shown that\n\\begin{prop}\n\n\n\\label{prop_non_deg2}\n\nFor type II  Einstein spacetimes, {the existence of} a non-twisting, non-degenerate double WAND implies that the algebraic type is necessarily II(d), i.e. $\\Phia_{ij}=0$, or more special, and\n\n\\begin{enumerate}\n\n\\item if $\\Phi_{ij}\\neq 0$ the spacetime is shearfree (Robinson-Trautman), i.e. the eigenvalue structure of $\\rho_{ij}$ is \n $\\{a,\\ldots,a\\}$ {(with $a\\neq0$)}. This is {the only possibility when} $n=4,5$.\\footnote{Since type II(abd) coincides with type III in those dimensions.} \nOne has $\\Phi_{ij}=A^0\\delta_{ij}\/(r-b_0)^{n-1}$, so that the type is D(bd) \n(one can set $b_0=0$ by a shift of the affine parameter $r$).\n\n\n\\item if $\\Phi_{ij}=0$ {(type II(abd))} the structure is  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special; $a,b,c_\\alpha\\neq0$). In particular, in six dimensions this means \n$\\{a,a,b,b\\}$ (see also section~\\ref{sec_6D}).\n\n\n\\end{enumerate}\n\\end{prop}\nThis is nothing but a more detailed version of points (\\ref{RT}) and (in the subcase $\\det\\rhob\\neq0$) (\\ref{Phi0}) of theorem~\\ref{prop_GSHD} of section~\\ref{sec_intro}, which are thus proven (the proof of  (\\ref{Phi0}) will be completed in section~\\ref{sec_nontwist} for any value of rank($\\rhob$), cf. remark~\\ref{rem_Phi0}).\nIn particular, (for $n>5$) there are always at least two double eigenvalues. \nIn this sense, we will see that the situation is different in the degenerate case. {Note that spacetimes of point 1. (all explicitly known \\cite{PodOrt06}) obey the optical constraint, whereas those of point 2. do not, in general. Examples of the latter in $n\\ge7$ dimensions will be provided in appendix~\\ref{app_violating}.}\n\n\n\n\n\n\\section{Non-twisting $\\bl$: degenerate $\\rhob$ ($0<m<n-2$) }\n\n\\label{sec_nontwist}\n\nLet $m$ denote the rank of $\\rho_{ij}$. The value $m=n-2$ corresponds to the previously considered non-degenerate case, while $m=0$ defines Kundt. Therefore here we shall restrict to $0<m<n-2$ .\nWe need to define two types of indices to distinguish between non-vanishing and vanishing eigenvalues, namely\n\\beqn\n & & \\rho_p=\\frac{1}{r-b_p} \\qquad (p,q,o,t=2,\\ldots, m+1) , \\\\\n & & \\rho_z=0 \\qquad (z,v,w,y=m+2,\\ldots, n-1) . \\label{rho_z_0}\n\\eeqn\n\nRecall that (point (i) of theorem~\\ref{prop_GSHD}) \nin the non-twisting case with $\\Phi^A_{ij}\\neq0$ one has $m=0,2$ and $\\rho_{ij}=\\mbox{diag}(a ,a ,0,\\ldots,0)$ for any $n>4$, {so that this case does not require further investigation}. \nIn all remaining cases we thus have \n\\be\n \\Phi^A_{ij}=0 \\qquad (m\\neq 0,2).\n\\ee\n{In the following we will give the $r$-dependence of the Weyl components, which is then used to constraint the possible forms of $\\rhob$. In particular, we shall explore under what conditions  all the eigenvalues of $\\rhob$ {can} be distinct (which is not permitted in the non-degenerate case). We shall also discuss some special cases and construct explicit examples.}\n\n\n\\subsection{$\\Phi_{ij}$ and $W_{ij}$ components}\n\n\\label{subsec_Phi_W_deg}\n\nProceeding similarly as in section~\\ref{sec_nondeg} but {(thanks to (\\ref{rho_z_0}))} with the additional equation $D\\Phi_z=0$, one finds\n\\beqn\n & & \\Phi_{p}=\\rho_p(A+\\Phi^0_{(p)}) , \\qquad \\Phi_{z}=\\Phi_{z}^0 , \\\\\n & & W_{pq}=\\frac{1}{(r-b_{(p)})(r-b_{(q)})}\\left[2\\int\\d r A+(\\Phi^0_{p}+\\Phi^0_{q})r+ W^0_{pq}\\right] , \\\\\n & & W_{pz}=\\frac{1}{r-b_{(p)}}(\\Phi^0_zr+W_{pz}^0) , \\qquad W_{zv}=W_{zv}^0 ,\n\n\\eeqn\nwhere \n\\be\n A=\\left(\\prod_o\\frac{1}{r-b_o}\\right)\\left[-\\sum_p\\Phi^0_{p}\\int\\d r\\prod_{q\\neq p}(r-b_q)-\\sum_z\\Phi^0_z\\int\\d r\\prod_{q}(r-b_q)+A^0\\right] .\n \\label{A_degen}\n\\ee\n\n\nNow, imposing $-2\\Phi_z=\\sum_pW_{zp}+\\sum_vW_{zv}$ we obtain the constraints\n\\be\n \\sum_vW_{zv}^0=-(m+2)\\Phi^0_z , \\qquad \\sum_p\\frac{\\Phi^0_zb_p+W^0_{pz}}{r-b_p}=0 .\n \\label{constr_Phi_z}\n\\ee\n\n\nNext, imposing $-2\\Phi_q=\\sum_pW_{qp}+\\sum_zW_{qz}$ gives \n\\be\n -2(A+\\Phi^0_{q})=\\sum_{p\\neq q}\\frac{1}{r-b_{(p)}}\\left[2\\int\\d r A+(\\Phi^0_{q}+\\Phi^0_{p})r+ W^0_{pq}\\right]+\\sum_z(\\Phi^0_zr+W^0_{qz}) .\n \\label{constrW_deg}\n\\ee\nBy comparing the leading terms of the l.h.s. and of the r.h.s. for $r\\to\\infty$ one obtains \n\\be\n m\\Phi^0_{q}+\\frac{1}{m+1}b_q\\sum_z\\Phi^0_z+\\sum_zW^0_{qz}=\\sum_p\\Phi^0_{p}-\\frac{1}{m+1}\\sum_pb_p\\sum_z\\Phi^0_z .\n \\label{constrW_deg_leading}\n\\ee\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ components} \n\n\\label{subsec_Phi3_4_deg}\n\nTo complete the description of the Weyl tensor, we now give the general form of the $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ components (recall that these are non-zero only for $n\\ge6$). In particular, we also discuss constraints following from the assumption that all  non-vanishing eigenvalues of $\\rho_{ij}$ are distinct (useful for later analysis). \n\n\\subsubsection{$\\Phi^{\\{3\\}}_{ijkl}$ components}\n\\label{subsubPhi3deg} \n\nEq.~\\eqref{B4}  gives\n\\be\n \\Phi_{(o)(p)(o)(q)}(\\rho_{(q)}-\\rho_{(p)})=0=\\Phi_{(z)(p)(z)(q)}(\\rho_{(q)}-\\rho_{(p)}) , \\qquad \\Phi_{(q)p(q)z}=0=\\Phi_{(v)p(v)z} , \\quad (p\\neq q, \\ v\\neq z) .\n \\label{A12_deg_Phi3}\n\\ee\nNon-vanishing components must satisfy the tracefree conditions (recall that $\\Phi_{ij}$ is diagonal in the frame we are using)\n\\be\n \\Phi_{poqo}+\\Phi_{pzqz}=0 , \\qquad \\Phi_{zpvp}+\\Phi_{zwvw}=0 \\qquad (p\\neq q, \\ v\\neq z) .\n \\label{trace_deg_Phi3}\n\\ee\n\nThe $r$-dependence, following from~\\eqref{Bi3}, is\n\\beqn\n & & \\Phi_{(p)z(p)v}=\\frac{\\Phi_{(p)z(p)v}^0}{r-b_{(p)}} \\quad (v\\neq z), \\qquad \\Phi_{(z)v(z)w}={\\Phi_{(z)v(z)w}^0} \\quad (v\\neq w) , \\nonumber \\\\\n & & \\Phi_{(p)q(p)o}=\\frac{\\Phi_{(p)q(p)(o)}^0}{(r-b_{(p)})(r-b_{(o)})} ,  \\quad (q\\neq o) \\qquad  \\Phi_{(z)p(z)q}=\\frac{\\Phi_{(z)p(z)(q)}^0}{r-b_{(q)}}  \\quad (p\\neq q) .\n \\label{r_deg_Phi3}\n\\eeqn\nFrom~(\\ref{trace_deg_Phi3}), the quantities in~(\\ref{r_deg_Phi3}) must satisfy the constraints\n\\beqn\n & & \\Phi_{pzqz}^0=0 , \\qquad \\sum_o\\frac{\\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \\qquad (p\\neq q) , \\label{r_deg_Phi3_constr_0}\\\\\n & & \\Phi^0_{zwvw}=0 , \\qquad \\sum_p\\frac{\\Phi_{z(p)v(p)}^0}{r-b_{(p)}}=0 \\qquad (v\\neq z) . \\label{r_deg_Phi3_constr}\n\\eeqn\n\n\nNote that, by the symmetries of the Weyl tensor, {from} the {third} equation of~(\\ref{r_deg_Phi3}) it follows that \n$\\Phi_{(p)q(p)o}\\neq0\\Rightarrow b_q=b_o$ (for $q\\neq o$), in agreement with~(\\ref{A12_deg_Phi3}); using also the {second} equation of~\\eqref{r_deg_Phi3_constr_0} we get\n\\be\n \\Phi^{\\{3\\}}_{(p)q(p)o}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0). \n \\label{Phi3_new}\n\\ee\nFurther, {the fourth} equation of~(\\ref{r_deg_Phi3}) gives $\\Phi_{(z)p(z)q}\\neq0\\Rightarrow b_q=b_p$ {(for $q\\neq p$)}, i.e., the structure is $\\{a,a,c_1\\dots\\}$ with $a\\neq0$, in agreement with~(\\ref{A12_deg_Phi3}). For $\\Phi_{z(p)v(p)}\\neq0$, {the second equation of}~\\eqref{r_deg_Phi3_constr} also implies $\\{a,a,c_1\\dots\\}$ with $a\\neq0$. \n\n\nWe are interested, in particular, in determining what are the necessary conditions in order to have all the non-vanishing eigenvalues distinct. From the {above} observations it follows that one necessarily has $\\Phi_{(p)q(p)o}=\\Phi_{(z)p(z)q}=\\Phi_{z(p)v(p)}=0$, therefore the only non-zero $\\Phi^{\\{3\\}}_{ijkl}$ components can be the $\\Phi_{z(w)v(w)}$, however with the constraint $\\Phi^0_{zwvw}=0$ (eq.~(\\ref{r_deg_Phi3_constr})).  \nIt is easy to see that for this to be satisfied in a non-trivial way,  indices $z,v\\ldots$  must run at least over four values, i.e. there must be at least four zero eigenvalues of $\\rho_{ij}$ (excluding Kundt, the spacetime must thus be at least seven-dimensional). In other words:\n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0$. \n\\end{itemize}\n\n\n\n\\subsubsection{$\\Phi^{\\{4\\}}_{ijkl}$ components}\n\\label{subsubPhi4deg} \n\nFrom~\\eqref{Bi3} and the symmetries of the Weyl tensor one finds\n\\beqn\n & & \\Phi_{pqot}=\\frac{\\Phi_{pq(o)(t)}^0}{(r-b_{(o)})(r-b_{(t)})} , \\qquad \\Phi_{pzqw}=\\frac{\\Phi_{pz(q)w}^0}{r-b_{(q)}}  , \\qquad \\Phi_{zvwy}=\\Phi_{zvwy}^0 , \\\\\n & & \\Phi_{pqoz}=0 , \\qquad \\Phi_{pqzw}=0, \\qquad \\Phi_{pzvw}=0 .\n \\label{r_deg_Phi4}\n\\eeqn\nSimilarly as in the case of $\\Phi^{\\{3\\}}_{ijkl}$, we observe that \n{if} $\\Phi_{pqot}\\neq0$ then $b_t=b_q$ and $b_o=b_p$ (or $b_t=b_p$ and $b_o=b_q$), i.e., \n\\be\n \\Phi^{\\{4\\}}_{pqot}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0). \n \\label{Phi4_new}  \n\\ee\n\n\n{Additionally,} {if} $\\Phi_{pzqw}\\neq0$ then\n$b_q=b_p$, {so that the structure is} { $\\{a,a,c_1\\dots\\}$ with  $a\\neq0$}. \n\n \nHaving  all the non-vanishing eigenvalues distinct thus requires $\\Phi_{pqot}=0=\\Phi_{pzqw}$ and the only non-zero components can be  $\\Phi_{zvwy}$. Therefore we conclude again that  indices $z,v\\ldots$  must run at least over four values unless $\\Phi^{\\{4\\}}_{ijkl}=0$ (and, again, excluding Kundt, the spacetime must thus be at least seven-dimensional), i.e., \n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $\\Phi^{\\{4\\}}_{ijkl}=0$. \n\\end{itemize}\n\n\n\\subsection{Case $\\Phi_{ij}=0$ (type II(abd))}\n\n\\label{subsec_Phi=0}\n\nThis case (non-trivial only for $n>5$) is obtained by setting\n\\be\n A=-\\Phi^0_p=K^0 , \\qquad \\Phi^0_z=0 \n\\ee\n{in the results obtained in~\\ref{subsec_Phi_W_deg}.}\nFor the $W_{ij}$ components one thus has\n\\be\n W_{pq}=\\frac{W^0_{pq}}{(r-b_{(p)})(r-b_{(q)})} , \\qquad W_{pz}=\\frac{W_{(p)z}^0}{r-b_{(p)}} , \\qquad W_{zv}=W_{zv}^0 ,\\label{Phi=0_Wpq}\n\\ee\nwith\n\\be\n \\sum_vW_{zv}^0=0 , \\qquad \\sum_p\\frac{W^0_{(p)z}}{r-b_{(p)}}=0 , \\qquad \\sum_zW^0_{qz}=0 , \\qquad \\sum_{p}\\frac{W^0_{(p)q}}{r-b_{(p)}}=0 .\n \\label{Phi=0_constr}\n\\ee\n\nIn view of these constraints (and of the properties of $W_{ij}$) we can briefly comment on some special cases:\n\n\\begin{itemize}\n \\item $\\Phi_{ij}=0$, $m=1$ $\\Rightarrow$ $W_{pq}=0=W_{pz}$, \n \\item $\\Phi_{ij}=0$, $m=2$ or $m=3$ $\\Rightarrow$ $W_{pq}=0$, \n \\item $\\Phi_{ij}=0$, $m=n-3$ $\\Rightarrow$ $W_{zv}=0=W_{pz}$, \n \\item $\\Phi_{ij}=0$, $m=n-4$ or $m=n-5$ $\\Rightarrow$ $W_{zv}=0$. \n\\end{itemize}\n\nFor special values of $n$ and $m$ some of these can hold simultaneously, thus leading to $W_{ij}=0$. Recalling the trivial implications $m=0\\Rightarrow W_{pq}=0=W_{pz}$ and $m=n-2\\Rightarrow W_{zv}=0=W_{pz}$ (valid also for $\\Phi_{ij}\\neq 0$), we have in particular \n\\begin{itemize}\n \\item for $n=4$ and $m=0,1,2$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$, \n \\item for $n=5$ and $m=0,1,2,3$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$, \n \\item for $n=6$ and $m=1,3$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$. \n\\end{itemize}\nWhile the first two implications simply reproduce the known result that $\\Phi_{ij}=0\\Leftrightarrow W_{ij}=0$ for $n=4,5$ (for any permitted $m$), the last remark will be useful for later purposes.\n\n\n\n\nUsing \\eqref{Phi=0_Wpq}, \\eqref{Phi=0_constr} and a reasoning similar to that of section \\ref{subsec_onerep} (in the paragraph after eq.~\\eqref{Wij}, including footnote~\\ref{foot_311}) one can also show that\n\\be\n W_{pq}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0),  \n \\label{W_new}\n\\ee\nwhich will be useful in the following.\n\n\n\\subsubsection{No repeated non-zero eigenvalues}\n\n\\label{subsubsec_Phi=0_norepeated}\n\nIn case all the $b_p$ are distinct, {equations~(\\ref{Phi=0_constr})} imply\n\\be\n W^0_{pz}=0=W^0_{pq} ,\n\\ee\nso that the only non-zero components of $W_{ij}$ are the $W_{zv}=W_{zv}^0$, with $\\sum_vW_{zv}^0=0$. This implies that, in order to have $W_{ij}\\neq0$, one needs that  indices $z,v,\\ldots$ run at least over  four values, i.e., the cases $m=n-3$, $m=n-4$ and $m=n-5$ imply $W_{ij}=0$. Thus we have proven: $\\Phi_{ij}=0$, all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $W_{ij}=0$. {Together with the results of subsection~\\ref{subsec_Phi3_4_deg}, we see that if such assumptions hold all boost weight zero components must actually vanish, so that we can conclude}\n\n\\begin{prop} \n\\label{prop_IIabd}\nFor type II(abd) ($\\Phi_{ij}=0$) non-twisting Einstein spacetimes with (degenerate) $\\rhob$ of rank $n-6<m<n-2$, the eigenvalue structure of $\\rhob$ is \n$\\{a,a, c_1\\ldots, 0,\\ldots\\}$ (with $a\\neq0$) or more special {(i.e., at least two non-zero eigenvalues of $\\rhob$ coincide)}.\n\\label{prop_Phi=0}\n\\end{prop}\n\n\n\n\nThe condition on $m$ means that either $m=n-3$ or $m=n-4$ or $m=n-5$. We will see {in section~\\ref{subsec_n-3}} below that in the case $m=n-3$ the assumption $\\Phi_{ij}=0$ can in fact be dropped {(since $\\Phi_{ij}\\neq0$ is not possible if all non-zero eigenvalues are distinct).  For $m=n-4$ the eigenvalue structure is thus $\\{a,a,0,0,c_1,\\ldots,c_{n-6}\\}$ and for $m=n-5$ it is $\\{a,a,0,0,0,c_1,\\ldots,c_{n-7}\\}$ (while for any smaller $m$ there are, of course, $n-2-m\\ge 4$ vanishing eigenvalues).} \nRecall also that for $m=n-2$ (non-degenerate case) we had a similar {result, cf.~proposition~\\ref{prop_non_deg2}.}\n\n\n\n\n\n\\subsection{The special case $m=n-3$}\n\n\\label{subsec_n-3}\n\n\n\nFor $n=4$ this case is not possible since it gives $m=1$ (see section~\\ref{subsec_m=1}). \n{In higher dimensions, the simplest spacetimes of this class are obtained by taking a Brinkmann warp {\\cite{Brinkmann25} (see also \\cite{OrtPraPra11})} of a type II Einstein spacetime possessing a non-degenerate mWAND with a single (spacelike) extra-dimension (cf. also \\cite{OrtPraPra11}). They include, e.g., {static} black strings and} this case may thus be of special interest. \n\nSince there is only one vanishing eigenvalue of $\\rho_{ij}$, i.e., $z$ can take only one value, we have $W_{zv}=0$ and, from~(\\ref{constr_Phi_z}),\n\\be\n \\Phi^0_z=0 , \\qquad \\sum_p\\frac{W^0_{pz}}{r-b_p}=0 .\n \\label{constr_m=n-3}\n\\ee\n\n\nTherefore (\\ref{constrW_deg_leading}) reduces to\n\\be\n (n-3)\\Phi^0_{q}+W^0_{qz}=\\sum_p\\Phi^0_{p} .\n\\ee\n\nFrom the results of subsection~\\ref{subsec_Phi3_4_deg}, when $m=n-3$ the only possible non-zero components of $\\Phi^{\\{3\\}}_{ijkl}$ can be $\\Phi_{(p)q(p)o}$, with $q\\neq o$ (in particular, $\\Phi_{pzqz}=\\Phi_{p(z)q(z)}=0$,  see the first of \\eqref{r_deg_Phi3_constr_0}), which satisfy\n\\be\n \\sum_o\\frac{\\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \\qquad (p\\neq q) , \n\\ee\nwhile the only possible non-zero components of $\\Phi^{\\{4\\}}_{ijkl}$ are $\\Phi_{pqot}$ ($\\Phi_{pqoz}=0$, see the first of \\eqref{r_deg_Phi4}) . \nFrom sections \\ref{subsubPhi3deg} and \\ref{subsubPhi4deg} (see \\eqref{Phi3_new}, \\eqref{Phi4_new})     \nit thus follows that if $\\Phi^{\\{3\\}}_{ijkl}\\neq0$ ($\\Leftrightarrow\\Phi^{\\{3\\}}_{(p)q(p)o}\\neq0$ here) or $\\Phi^{\\{4\\}}_{ijkl}\\neq0$ ($\\Leftrightarrow\\Phi^{\\{4\\}}_{pqot}\\neq0$ here)  then the eigenvalue structure is  $\\{a,a,b,b,c_1\\dots\\}$, where  $a,b\\neq0$. Moreover, it is easy to see that when $m=n-3$ both $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$  can be non-zero only for $n\\ge7$.  This is of interest only in six dimensions ($\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ identically for $n=4,5$) and can be summarized as \n\\begin{itemize}\n \\item $n=6$, $m=3$  $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$. \n\\end{itemize}\n\nIf we additionally assume $\\Phi_{ij}=0$ (then also $W_{zv}=0=W_{pz}$, see comments after \\eqref{Phi=0_constr}), then some of the components $W_{pq}$, $\\Phi^{\\{3\\}}_{p(o)q(o)}$ or $\\Phi^{\\{4\\}}_{pqot}$ have to be non-vanishing (otherwise the spacetime would be of type III). Then using also results of section~\\ref{subsec_Phi=0}  {(eq.~\\eqref{W_new})} it follows that type II spacetimes {with $m=n-3$ and $\\Phi_{ij}=0$} {must have} the eigenvalue structure $\\{a,a,b,b,c_1\\dots c_{n-7},0\\}$. \n\n\n\n\n\\subsubsection{No repeated non-zero eigenvalues}\n\n\\label{subsubsec_m=n-3_no_rep}\n\nIf all $b_p$ are distinct from the second of~(\\ref{constr_m=n-3}) we obtain  \n\\be\n W^0_{pz}=0 ,\n\\ee\nso that $W_{pz}=0$ and all the $\\Phi^0_{q}$ coincide, namely, \n\\be\n \\Phi^0_{q}=\\frac{1}{(n-3)}\\sum_p\\Phi^0_{p}\\equiv f^0 .\n\\ee\nTherefore\n\\be\n A=-f^0+A^0\\prod_p\\frac{1}{r-b_p} ,\n\\ee\nand one can basically proceed as in the non-degenerate case. Comparing the structures of the poles of the l.h.s. and r.h.s. of (\\ref{constrW_deg}) one again arrives at\n\\be\n A^0=0, \\qquad W^0_{pq}=0 .\n\\ee\nCombining this with the previous results we conclude $\\Phi_q=0$, $\\Phi_z=0$, $W_{qz}=0$, $W_{qp}=0$, $W_{zv}=0$, i.e.\n\\be\n \\Phi_{ij}=0=W_{ij} .\n\\ee\nSince here $m=n-3>n-6$, we can use the results of subsection~\\ref{subsec_Phi3_4_deg} to conclude that also $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$, so that all b.w. 0 components of the Weyl tensor vanish. Thus we have proven: {\\it all non-zero $\\rho_{ij}$ are distinct and $m=n-3$ $\\Rightarrow$ all b.w. 0 components vanish.}\nIn other words, {\\it in the case $m=n-3$, the eigenvalue structure of $\\rhob$ must be $\\{a,a,c_1, \\dots ,c_{n-5}, 0\\}$ \n(or more special)}. If, additionally, $\\Phi_{ij}=0$, there is in fact another repeated non-zero eigenvalue \\cite{Wylleman_priv}\\footnote{We thank Lode Wylleman for pointing this out.} {(see the comment just before section \\ref{subsubsec_m=n-3_no_rep}), so that} \n Proposition~\\ref{prop_Phi=0} can accordingly be reformulated as (see also \\cite{Wylleman_priv}) \n \n\\begin{prop} \n\\label{prop_n-3_text}\nIn a type II Einstein spacetime of dimension $n>4$ with a non-twisting multiple WAND of rank $m=n-3$, the eigenvalue structure of $\\rhob$ is $\\{a,a,c_1,\\ldots c_{n-5},0\\}$ where $a,c_\\alpha  \\ne 0$ (and necessarily $\\Phi^A_{ij}=0$ if $n>5$). If, additionally, $\\Phi_{ij}=0$, the structure becomes $\\{a,a,b,b,c_1,\\ldots c_{n-7},0\\}$ where $a,b,c_\\alpha  \\ne 0$.\n\\end{prop}\n\n\n\n\\begin{rem}\n\\label{rem_Phi0}\n{Together with propositions~\\ref{prop_non_deg2} and \\ref{prop_Phi=0}, this shows that, for any possible value of $m$ ($0<m\\le n-2$), the mWAND of type II(abd) (i.e., $\\Phi_{ij}=0$) Einstein spacetimes has  $\\rhob$ of the form $\\{a,a,b,b,c_1,\\ldots c_{n-6}\\}$ (where $a\\ne 0$ if $m=n-5$ or $m=n-4$ and $a,b\\ne 0$ if $m=n-3$ or $m=n-2$ ).} In particular, point~(\\ref{Phi0}) of theorem~\\ref{prop_GSHD} is thus now also proven.\n\\end{rem}\n\n\n\n\n\n\n\n\\begin{rem}\n\\label{rem_5d}\nIn five dimensions (for which always $\\Phi_{ij}\\neq0$), recalling also the results for the non-degenerate case {(proposition~\\ref{prop_non_deg2})}, this enables us to list all the admitted eigenvalues structures, {for $m=3,2,1,0$, respectively}: $\\rho_{ij}=\\mbox{diag}(a,a,a)$,\n$\\rho_{ij}=\\mbox{diag}(a,a,0)$,  \n$\\rho_{ij}=\\mbox{diag}(a,0,0)$,  \nor $\\rho_{ij}=0$ (Kundt). See \\cite{Ortaggioetal12} for more details and examples. \n\\end{rem}\n\n\\begin{rem}\n\\label{rem_6d}\nIn six dimensions, we conclude that the case $m=3$ requires $\\rho_{ij}=\\mbox{diag}(a,a,b,0)$. However, as shown above in this case one has $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and additionally the implication $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ holds (see subsection~\\ref{subsec_Phi=0}). Therefore one necessarily has $\\Phi_{ij}\\neq0$ when $n=6$ and $m=3$ (or all Weyl boost zero components would vanish). {See section~\\ref{sec_6D} for more details.}\n\\end{rem}\n\n\n\n\n\n\n\\subsection{The special case $m=1$} \n\n\\label{subsec_m=1}\n\n\n\n\n\nFrom a viewpoint complementary to that of the previous subsection, it is also interesting to analyze the case when only one eigenvalue is non-zero. Let us associate the index $p=2$ to this eigenvalue, while $z,v=3,\\ldots,n-1$. Using the general equations presented above, one obtains (cf. also~\\cite{PraPra08}) \n\\be\n \\Phi_z=\\Phi^0_z , \\qquad \\Phi_2=-\\frac{1}{2}\\sum_z\\Phi^0_z, \\qquad W_{2z}=\\Phi^0_z , \\qquad \\sum_vW^0_{zv}=-3\\Phi^0_z , \n\\ee\nwhile clearly here $W_{pq}=0$ (note that for $n=4$ one gets \n$\\Phi^0_z=0$  \nand $\\Phi_{ij}=0=W_{ij}$, so this case does not occur). Therefore all components of $\\Phi_{ij}$ and $W_{ij}$ are $r$-independent. Note that  $\\Phi_{ij}=0\\Leftrightarrow \\Phi^0_z=0$. When this happens the only non-zero components of $W_{ij}$ can be the $W^0_{zv}$, but with the constraint $\\sum_vW^0_{zv}=0$. Therefore it is easy to see that $W_{ij}$ can be non-zero only if $z,v$ can take at least four values, i.e., $z,v=3,4,5,6,\\ldots$, so that $n\\ge 7$. To summarize \\begin{itemize}\n \\item $n<7$, $m=1$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ .\n\\end{itemize}\nThis can be understood as a special instance of the results of subsection~\\ref{subsec_Phi=0}. \nThis result is of interest in six dimensions, since one always has $\\Phi_{ij}=0\\Rightarrow W_{ij}=0$ in four and five dimensions. Namely, for $n=6$ the case $\\{a ,0,0,0\\}$ is forbidden if $\\Phi_{ij}=0\\neq W_{ij}$. \n\nFrom the results of subsection~\\ref{subsec_Phi3_4_deg}, when $m=1$ the only possible non-zero components of $\\Phi^{\\{3\\}}_{ijkl}$ can be $\\Phi_{(z)v(z)w}$ (with $v\\neq w$), which satisfy\n\\be\n \\Phi^0_{zvzw}=0 \\qquad (v\\neq w) , \n\\ee\nwhile the only possible non-zero components of $\\Phi^{\\{4\\}}_{ijkl}$ are $\\Phi_{zvwy}$. It is easy to see that, since $m=1$,  both sets of components vanish identically unless $n\\ge7$, so that \n\\begin{itemize}\n \\item $n=6$, $m=1$  $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$. \n\\end{itemize}\n\n\\begin{rem}\n\\label{rem_6d_m=1}\nSince for $n=6$, $m=1$ one has $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ (see above), we conclude that in six dimensions spacetimes with $m=1$ require $\\Phi_{ij}\\neq 0$.\n\\end{rem}\n\n\n\\subsubsection{Examples}\n\n\\label{subsubsec_m=1}\n\nFor any $n\\ge5$, simple examples of spacetimes with $m=1$ {(and thus $\\Phi^A_{ij}=0$)} are given by:\n\\begin{enumerate}\n \\item dS$_3\\times$S$_{n-3}$ (or AdS$_3\\times$H$_{n-3}$): $n\\ge5$, $\\Phi_{ij}\\neq 0$, $\\Lambda\\neq 0$, \\label{dSxS_m=1}\n \\item as above but with a Brinkmann warp: $n\\ge6$, $\\Phi_{ij}\\neq 0$, also $\\Lambda=0$ is possible, \\label{Brink_m=1} \n \\item Minkowski$_3\\times$(Ricci-flat)$_{n-3}$: $n\\ge7$, $\\Phi_{ij}=0$, $\\Lambda=0$. \\label{MinkxRc_m=1}\n\\end{enumerate}\n\nIn the first case, if {we restrict to} $n\\ge7$ the second factor space can be in fact {\\em any} Einstein space (with Ricci scalar given by $2(n-3)K$). The metric of the first and third case can thus be given in a unified way as\n\\be\n \\d s^2 =\\left(1+\\frac{K}2ur\\right)^{-2}(2\\d u\\d r+r^2\\d x^2)+\\d\\Sigma^2 ,\n\\ee\nwhere $\\d\\Sigma^2$ is the metric of an $(n-3)$-dimensional Einstein space with Ricci scalar $R_\\Sigma=2(n-3)K$ and $K$ is a constant related to the $n$-dimensional Ricci scalar by $R=2nK$. A geodesic twistfree mWAND $\\bl$ (degenerate but expanding along $x$) is\n\\be\n {\\ell}_a\\d x^a=\\d u .\n\\ee\nNote that here $r$ is an affine parameter along $\\bl$ only in the case $K=0$. {Such product spaces are of type D \\cite{PraPraOrt07} and in fact any null vector field tangent to the three-dimensional Lorentzian factor is an mWAND \\cite{GodRea09,PraPraOrt07}.}\nUsing these results metrics for the second case can also be constructed straightforwardly. {These will be also of type D \\cite{OrtPraPra11}.}\n\n\n\n\\subsection{Case when all non-zero eigenvalues are distinct} \n\n\\label{subsec_alldistinct}\n\n\n\nBy combining the results of subsections~\\ref{subsec_Phi=0} and \\ref{subsec_Phi3_4_deg} we have \n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct, $m=n-4$ or $m=n-5$ and $\\Phi_{ij}=0$ $\\Rightarrow$ all b.w. 0 components vanish,\n \\item all non-zero $\\rho_{ij}$ are distinct and $m=n-3$ $\\Rightarrow$ all b.w. 0 components vanish.\\footnote{By adding the results of \\cite{Pravdaetal04} we can in fact conclude that in both cases the whole Weyl tensor must vanish.} \n\\end{itemize}\nTherefore type II Einstein spacetimes for which all non-zero $\\rho_{ij}$ are distinct require that at least one of the following holds ({obviously $m>0$ or we would simply have Kundt}):\n\\begin{enumerate}\n \\item $1\\le m\\le n-6$ (i.e., at least four eigenvalues of $\\rho_{ij}$ vanish),\n \\item $\\Phi_{ij}\\neq 0$ and $1\\le m\\le n-4$ (i.e., at least two eigenvalues of $\\rho_{ij}$ vanish).\n\\end{enumerate}\nThe first case can occur only for $n\\ge7$: for $n=7$ one can have only $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0)$, for $n=8$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0,0)$ or  $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0,0,0)$, etc.. The second case can occur for $n\\ge5$: for $n=5$ one has simply $\\rho_{ij}=\\mbox{diag}(a ,0,0)$, for $n=6$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0)$, for $n=7$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,c ,0,0,0)$, etc.. These results are summarized in table~\\ref{tab_distinct_eigenval}. For such solutions {\\em the optical constraint is clearly violated if $m>1$}. The {previously discussed} case $m=1$ has been included for completeness but it is of course ``trivial'' in this context since there is a single non-zero eigenvalue.\n\n\n\\begin{table}[t]\n \\begin{center}\n  \\begin{tabular}{|c|c|l|c|}\n   \\hline Case & $n$ & Possible $m$ & Examples  \\\\\\hline\n    $\\Phi_{ij}\\not=0$ & 5 & 1 & \\\\\n    \t\t\t\t\t   \t\t  & 6 & 1,2 &   \\\\\n     \t\t\t\t\t\t\t\t\t& 7 & 1,2,3 &  \\\\\n     \t\t\t\t\t\t\t \t\t& 8 & 1,2,3,4 &  \\\\\n     \t\t\t\t\t\t\t \t\t& \\vdots &  &  \\\\\n     \t\t\t\t\t\t\t\t \t& $n$ & 1,2,3\\ldots, $n-4$ & (A)dS$_{m+2}\\times$S$_{n-2-m}$(H$_{n-2-m}$) \\\\\\hline\n     $\\Phi_{ij} =0$   & 7 & 1 &  \\\\\n        \t\t\t\t\t\t\t& 8 & 1,2 &  \\\\\n        \t\t\t\t\t\t  & \\vdots &  &  \\\\\n        \t\t\t\t\t\t\t& $n$ & 1,2,3,\\ldots, $n-6$ &  Mink$_{m+2}\\times$(Ricci-flat)$_{n-2-m}$  \\\\\n\\hline\n  \\end{tabular}\n    \\caption{Non-twisting Einstein spacetimes for which there are no repeated non-zero eigenvalues of $\\rho_{ij}$: permitted values of the rank $m$ of $\\rho_{ij}$ in various dimensions {(necessarily $1\\le m \\le n-4$ thanks to propositions~\\ref{prop_non_deg2}  and \\ref{prop_n-3_text})}. {As discussed in the text, different examples can also be generated using Brinkmann's warp, or by replacing the S$_{n-2-m}$(H$_{n-2-m}$) factor by a more general Einstein space. {Note that here solutions with $m>1$ violate the optical constraint.\\label{tab_distinct_eigenval}}\n    }}\n  \\end{center}\n\\end{table}\n\n\n\n\\subsubsection{Examples}\n\n\\label{subsubsec_examples_distinct}\n\nThe case $n=4$ is not possible. Examples with $m=1$ have been already presented {in subsection~\\ref{subsec_m=1} and will not be discussed again here}. For $m\\ge2$ the $n=5$ case is also forbidden {(cf. Remark~\\ref{rem_5d} and \\cite{Ortaggioetal12})}. \nFor $n=6$ the only possibility is $m=2$ (with $\\Phi_{ij}\\neq0$) and one can construct explicit Einstein spacetimes by taking dS$_4\\times$S$^2$ or AdS$_4\\times$H$^2$. These can then be extended to any dimension $n\\ge7$ by a simple Brinkmann warp (and if one starts from dS$_4\\times$S$^2$ the cosmological constant of the resulting spacetime can be arbitrary, cf.~\\cite{OrtPraPra11}), so to have explicit examples of the type $\\rho_{ij}=\\mbox{diag}(a ,b ,0,\\ldots,0)$ (i.e., with $m=2$) for any $n>5$. \n\nHowever, according to the previous comments (see also table~\\ref{tab_distinct_eigenval}), for $n=7$ one can have $m=2,3$, for $n=8$ it is possible $m=2,3,4$ and so on: in general, and for any $n>5$, $2\\le m\\le n-4$ explicit metrics can be constructed similarly as in six dimensions by taking dS$_{m+2}\\times$S$_{n-2-m}$ or AdS$_{m+2}\\times$H$_{n-2-m}$ (where the two factor spaces must have {Ricci scalars} given by $(m+2)(m+1)K$ and $(n-2-m)(m+1)K$, respectively), and appropriate hypersurface orthogonal null congruences living in the (A)dS$_{m+2}$ factor {(explicit examples are given below)}. These have necessarily $\\Phi_{ij}\\neq0$ and $\\Lambda\\neq0$. Using Brinkmann's warp one can also generate Ricci-flat solutions (again with $\\Phi_{ij}\\neq0$) for which, however, the stronger restriction $n>6$, $2\\le m<n-4$ holds (at least if this method is used). Products of Ricci-flat spaces are possible if each factor is at least four-dimensional (therefore there are always at least four vanishing eigenvalues) and the Lorentzian factor is flat and fall into the $\\Phi_{ij}=0$ class, with $\\Lambda=0$ (these metrics belong to the type D \\pp waves mentioned in \\cite{OrtPraPra09}). \n\nSummary:\n\\begin{enumerate}\n \\item dS$_{m+2}\\times$S$_{n-2-m}$ (or AdS$_{m+2}\\times$H$_{n-2-m}$)\\footnote{Note that if $n-2-m\\ge4$ the second factor space can be in fact any Einstein space with Ricci scalar given by $(n-2-m)(m+1)K$.}: $n>5$, $2\\le m\\le n-4$, $\\Phi_{ij}\\neq 0$, $\\Lambda\\neq 0$, \\label{dSxS}\n \\item as above + Brinkmann, also $\\Lambda=0$: $n>6$, $2\\le m\\le n-5$, $\\Phi_{ij}\\neq 0$, \\label{Brink}\n \\item Minkowski$_{m+2}\\times$(Ricci-flat)$_{n-2-m}$: $n>7$, $2\\le m\\le n-6$, $\\Phi_{ij}=0$, $\\Lambda=0$. \\label{MinkxRc}\n\\end{enumerate}\n\nNote that all the above spacetimes {also} belong to the Kundt class, although w.r.t. a null congruence different from the one which is of interest for our discussion (which is expanding and shearing). {In fact they all admit $\\infty^m$ mWANDs:} similarly as in section~\\ref{subsubsec_m=1}, they are of type D (with $\\Phi^A_{ij}=0$) and any null vector field tangent to the Lorentzian factor is an mWAND -- so one can always also find an mWAND with an optical matrix having all non-zero eigenvalues identical.  {However, this mWAND ``degeneracy'' is not the generic situation} (see appendix~\\ref{app_violating} for an example of a class of type D spacetimes which admit (only) two double WANDs, both violating the optical constraint).\n\n\nExplicitly, for cases~\\ref{dSxS} and \\ref{MinkxRc} one can take, e.g., the metric\n\\be\n \\d s^2 =\\Omega^2(2\\d u\\d r+P_q^2\\d x_q^2)+\\d\\Sigma^2 ,\n \\label{ex_distinct}\n\\ee\nwhere $q=2,\\dots,m+1$, $\\d\\Sigma^2$ is the metric of an $(n-2-m)$-dimensional Einstein space with Ricci scalar $R_\\Sigma=(n-2-m)(m+1)K$ and\n\\be\n P_q=r-b_q, \\qquad \\Omega^{-1}=1+\\frac{K}{4}\\left[2r\\left(u-\\frac{1}{2}P_qx_q^2\\right)+P_q^2x_q^2\\right ] ,\n\\ee\n{in which $b_q$ and $K$ are constants.}\nThe first factor space in~(\\ref{ex_distinct}) is a space of constant curvature with Ricci scalar given by $(m+2)(m+1)K$, so that the Ricci scalar of the full spacetime is \n\\be\n R=n(m+1)K .\n\\ee\n\nThe geodesic, twistfree mWAND $\\bl$ and the remaining frame vectors are given by\n\\be\n {\\ell}_a\\d x^a=\\d u , \\qquad n_a\\d x^a=\\Omega^2\\d r, \\qquad m_{(q)a}\\d x^a=\\Omega P_{(q)}\\d x_{(q)} ,\n\\ee\nwhile vectors $m_{(v)a}$ will depend on the specific form of $\\d\\Sigma^2$. The optical matrix $\\rho_{ij}$ is given by\n\\be\n \\rho_{pq}=\\delta_{(p)q}\\frac{(\\Omega P_{(p)})_{,r}}{\\Omega^{3}P_{(p)}} , \\qquad \\rho_{pv}=0=\\rho_{vz} .\n\\ee\nIn the case $K=0$ (i.e., $\\Lambda=0$) one thus has $\\rho_{pq}=\\delta_{(p)q}\/(r-b_{(p)})$. Note, however, that $r$ is {\\em not} an affine parameter when $K\\neq 0$ (one has $\\bl=\\Omega^{-2}\\pa_r$). \n\n\nFor case~\\ref{Brink} one can use the method illustrated in detail in \\cite{OrtPraPra11}. \n\n\n\n\n\\section{Counterexamples}\n\n\\label{sec_counter}\n\n\nIn \\cite{Ortaggioetal12}, a counterexample to the converse of the five-dimensional ``shear-free'' part of the GS theorem was presented, thus demonstrating explicitly that the condition that the optical matrix $\\rhob$ admits a canonical form compatible with a geodetic mWAND is not sufficient for the null geodetic being  an mWAND. Similarly, also our results above give conditions that are (necessary but) not sufficient. In order to demonstrate that, here we present a few ``counterexamples'', i.e., certain Einstein spacetimes that admit a non-twisting null (thus geodesic) vector field $\\bl$ with $\\rhob$ taking one of the permitted ``canonical forms'' (cf. theorem~\\ref{prop_GSHD} and propositions~\\ref{prop_non_deg2}, \\ref{prop_IIabd} and \\ref{prop_n-3_text}) and yet with $\\bl$ not being an mWAND. Note that such counterexamples will necessarily be shearing: a twistfree shearfree null vector field is automatically an mWAND \\cite{PodOrt06,OrtPraPra07}. \n\n\n\n\\subsection{Non-degenerate $\\rhob=\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$}\n\nIn the non-degenerate case a ``counterexample'' to the canonical form $\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$ (with $a,b,c_\\alpha \\ne 0$ and $a\\neq b$ -- see points (ii) and (iii) of theorem~\\ref{prop_GSHD} and proposition~\\ref{prop_non_deg2}) is given by the following six-dimensional Ricci-flat spacetime, which belongs to a class of metrics considered by Robinson (as described in \\cite{Trautman02b}),\n\\be\n \\d s^2=2\\d u\\d r+2\\cosh^2r\\d w\\d\\bar w+2\\sin^2r\\d\\zeta\\d\\bar\\zeta .\n \\label{robinson}\n\\ee\nHere the hypersuface orthogonal null vector field $\\ell_a\\d x^a=\\d u$ is {\\em not} an mWAND (not even a single one) and yet the corresponding \n$\\rhob$ has the eigenvalue structure $\\{a,a,b,b\\}$ (with $a\\neq0\\neq b$, $a\\neq b$). Associated to $\\bl$ there is also an optical structure \\cite{Trautman02b}.\nNote, however, that spacetime~\\eqref{robinson} is a type N \\pp wave (albeit in {so-called Rosen coordinates, see, e.g., metric~(39) of \\cite{Brinkmann25}}), with a covariantly constant mWAND given by $n_a\\d x^a=\\d r$. \n\n\\subsection{Case $n-6<m<n-2$} \n\nDirect products of~\\eqref{robinson} with flat extra dimensions clearly provide non-WANDs with eigenvalues $\\{a,a,b,b,0,\\ldots,0 \\}$. In particular, for $n=7$ one obtains $\\{a,a,b,b,0\\}$, which is thus a counterexample to the $\\rhob$-form of Proposition~\\ref{prop_n-3_text}. For $n=8,9$ this is also a counterexample to the $\\rhob$-form of Proposition~\\ref{prop_IIabd}. Another counterexample to Proposition~\\ref{prop_n-3_text} can be obtained for $n=6$ by taking the direct product of a vacuum black ring \\cite{EmpRea02prl} with a flat dimension: in the coordinates of \\cite{ElvEmp03}, the hypersurface orthogonal null vector field  (also considered in 5d in \\cite{Taghavi-Chabert11} for different purposes)\n\\be\n \\ell_a\\d x^a = \\frac{\\sqrt{F(y)}}{G(y)}\\d y+\\d\\psi ,\n\\ee\ncorresponds to $\\rhob$ of the form $\\{a,a,b,0\\}$ and is not a WAND. \nAlternatively, the same form $\\{a,a,b,0\\}$ can be obtained by taking the direct product of a 5d vacuum static KK bubble \\cite{GodRea09} with a flat dimension (with a non-WAND $\\ell_a\\d x^a=\\d t+V(r)^{-1\/2}\\d r$ -- this 6d spacetime is of type D but the mWANDs are different from $\\bl$ \\cite{OrtPraPra11}).\n\n\n\\subsection{Case when all non-zero eigenvalues are distinct} \n\nExamples of spacetimes with a non-twisting, non-WAND $\\bl$ having all non-zero eigenvalues distinct can be constructed as a direct product of a 4d \\pp wave with flat space, i.e.,\n\\be\n \\d s^2=2\\d u\\d r+\\cosh^2r\\d x^2+\\sin^2r\\d y^2+\\d z_i\\d z^i .\n \\label{4d_pp}\n\\ee\nHere,  $\\ell_a\\d x^a=\\d u$ has a $\\rhob$ with eigenvalue structure $\\{a,b,0,\\ldots,0\\}$ (with $a\\neq0\\neq b$, $a\\neq b$). This is the form of $\\rhob$ discussed in section~\\ref{subsec_alldistinct} (see also table~\\ref{tab_6D} for the $n=6$ case). Similarly as the spacetime~\\eqref{robinson}, also \\eqref{4d_pp} is a type N \\pp wave with mWAND $n_a\\d x^a=\\d r$. \n\n\\subsection{Case $m=1$}\n\nExamples of non-twisting null congruences with $m=1$ (the case discussed in section~\\ref{subsec_m=1}) that are not multiple WANDs can be obtained from the (Ricci-flat) Newman-Tamburino lift considered in \\cite{Ortaggioetal12}, i.e.,\n\\be\n{\\rm d}s^2 = r^2 {\\rm d} x^2 + x^2 {\\rm d} y^2 - \\frac{4 r}{x } {\\rm d} u {\\rm d} x - 2 {\\rm d} u {\\rm d} r + x^{-2} \\left[c + \\ln (r^2 x^4)\\right] {\\rm d} u^2+\\d z_i\\d z^i , \\label{NTmetric}\n\\ee\nwhere $c$ is a constant. In this case $\\ell_a\\d x^a = {\\rm d} u$ is a {\\em single} WAND  with optical matrix of the form $\\{a,0,\\ldots,0\\}$ (with $a\\neq 0$). \n\n\n\n\n\n\\section{Existence of totally geodesic null two-surfaces}\n\n\\label{sec_totgeod}\n\nTheorem~\\ref{prop_GSHD} comprises the main results proven in the previous sections. In particular, it implies that there always exists at least a pair of repeated (possibly vanishing) eigenvalues and that (recall that $n\\ge 6$) there are in fact at least two such pairs when, e.g.,  $\\det\\rhob\\neq0$ (see also proposition~\\ref{prop_non_deg2}) or $\\Phi_{ij}=0$ (see also propositions~\\ref{prop_IIabd} and \\ref{prop_n-3_text}). The following proposition (see also the definitions in appendix~\\ref{app_shearfree}, in particular eq.~\\eqref{D}) elucidates the geometrical meaning of this fact (and provides a connection with the standard 4d version of the Goldberg-Sachs theorem, cf. appendix~\\ref{app_GS_4D}).\n\\begin{prop} \n In an algebraically special Einstein spacetime of dimension $n\\ge 6$ with a non-twisting {mWAND} $\\bl$, using a parallelly transported eigenframe of $\\rhob$\n \\begin{enumerate} \n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{a,a,c_1,\\ldots, c_{n-5},0 \\}$ with $c_\\alpha\\neq a\\neq 0$ then the totally null distribution ${\\cal D}_{23}$ is integrable with totally geodesic integral surfaces, \n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{0,0,c_1,\\ldots, c_{n-4}\\}$ with $c_\\alpha\\neq 0$ then the totally null distribution ${\\cal D}_{23}$ is integrable with totally geodesic integral surfaces,\n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$ with $b\\neq a$ and $a\\neq c_\\alpha\\neq b$ then the totally null distributions ${\\cal D}_{23}$ and ${\\cal D}_{45}$ are integrable with totally geodesic integral surfaces (either $a$ or $b$ can be zero). \n \\end{enumerate}\t\n\\label{prop_integrab}\n\\end{prop}\n\\begin{proof}\nThe proof is similar in all cases and relies on the use of the Ricci identity (11k,\\cite{OrtPraPra07}) (equivalent to the NP equation (A4,\\cite{Durkeeetal10})), which for algebraically special Einstein spacetimes reduces to \n\\be\n \\delta_{[j|} \\rho_{i|k]} =  L_{1[j|} \\rho_{i|k]}+ \\tau_i \\rho_{[jk]}+\\rho_{il}\\M{l}{[j}{k]}\n+ \\rho_{l[j|}\\M{l}{i|}{k]} , \\label{11k}\n\\ee\nwhere $\\delta_i=m_{(i)}^a\\nabla_a$ (recall also the definitions in \\eqref{L1i_M}). Since here $\\bl$ is twistfree we have $\\rho_{[jk]}=0$ in the above equation. Assume the form of $\\rhob$ specified in 1., i.e., $\\rho_{22}=\\rho_{33}=a\\neq 0$ (and all remaining eigenvalues different from $a$) and consider \\eqref{11k} for $i,j,k=23\\hat k\/32\\hat k$ (with $\\hat k\\neq 2,3$). This gives $\\M{2}{\\hat k}{3}=0=\\M{3}{\\hat k}{2}$. Then, \\eqref{11k} with $i,j,k=22\\hat k\/33\\hat k$ gives $\\M{2}{\\hat k}{2}=\\M{3}{\\hat k}{3}$,  which completes the proof of point 1. (see eqs.~\\eqref{2totgeod_real}). The proof of point 2. is identical. The proof of point 3. is analogous, after extending the reasoning to the index pair (45); in this case, however, either $a$ or $b$ can be zero, provided all $c_\\alpha\\neq 0$. \n\\end{proof}\n\\begin{rem}\nIn the case there are further pairs of repeated eigenvalues, the corresponding two-spaces will also define integrable totally geodesic distributions, provided these pairs are not repeated. In particular in even dimensions, if there are $(n-2)\/2$ (non-repeated) pairs of equal eigenvalues there will be $(n-2)\/2$ such totally geodesic two-surfaces.\n\\end{rem}\n\\begin{rem}\nPoint 1. of the above proposition includes, in particular, all non-twisting but expanding type III\/N Einstein spacetimes (for which $c_{\\alpha}=0$ \\cite{Pravdaetal04}) and type II non-twisting non-Kundt spacetimes with $\\Phi^A_{ij} \\ne 0$ (point \\eqref{PhiA} of theorem~\\ref{prop_GSHD}). \nPoint 3. is relevant to the case $\\det\\rhob\\neq0$ with $\\Phi_{ij}=0$ (proposition~\\ref{prop_non_deg2} and point \\eqref{Phi0} of theorem~\\ref{prop_GSHD}), which can only be of type II(abd).\n\\end{rem}\n\\begin{rem}\nThe proposition does not explicitly include the $n=5$ case. However, from Proposition~4 of \\cite{Ortaggioetal12} one can easily obtain the corresponding results (also in the presence of twist). In particular, in the non-twisting case ${\\cal D}_{23}$ is totally geodesic when the eigenvalue structure of $\\rhob$ is either $\\{a,a,0\\}$ or  $\\{0,0,a\\}$ (with $a\\neq 0$ in both cases). The same is true in the case $\\{a,a,a\\}\\neq0$, although this does not follow from (11k,\\cite{OrtPraPra07}) (instead, one can easily adapt the argument of footnote~10 of \\cite{Ortaggioetal12}). \n\\end{rem}\n\\begin{rem}\nRecall that in 4d the Goldberg-Sachs theorem implies that $\\bl$ is geodesic and the eigenvalue structure of $\\rho_{(ij)}$ is $\\{a,a\\}$, which is {\\em equivalent} to ${\\cal D}_{23}$ being integrable (and thus automatically totally geodesic, in 4d, see appendix~\\ref{app_GS_4D}).\n\\end{rem}\n\n\n\n\n\\section{Type II spacetimes with a non-twisting multiple WAND in six dimensions}\n\n\\label{sec_6D}\n\n\n\nAs already mentioned, five-dimensional algebraically special Einstein spacetimes have been studied in detail in \\cite{Ortaggioetal12} and, for $n=5$, the results obtained above are also contained in that reference. The next lower dimension to consider is thus $n=6$ and in this section our general results are specialized to this case. This is of interest also because of a qualitative difference between $n<6$ and $n\\ge 6$ dimensions: for $n<6$ type II spacetime necessarily have $\\Phi_{ij}\\neq 0$ (since type II coincides with type II(c) {for $n=4,5$}), while this is not so in higher dimensions.  \n\n\\subsection{Permitted forms of $\\rhob$}\n\n\n\\label{subsec_permitted_6D}\n\n\n\nOne has to consider the various cases $m=4,3,2,1$ ($m=0$ is Kundt).\n\nFirst, recall that for $m\\neq 2$ point \\eqref{PhiA} of theorem~\\ref{prop_GSHD} implies that $\\Phi^A_{ij}=0$. Let us thus first discuss this case.\n\nNow, for $m=4$ proposition~\\ref{prop_non_deg2} tells us that if $\\Phi_{ij}\\neq 0$ we are in the \\RT class, i.e. $\\rhob=\\mbox{diag}(a,a,a,a)\\neq 0$ (in which case $\\Phi_{ij}\\propto \\delta_{ij}$), otherwise we have $\\rhob=\\mbox{diag}(a,a,b,b)$ (with $a,b\\neq 0$).\n\n\n\nFor $m=3$ we learn from Proposition~\\ref{prop_n-3_text} that $\\rhob=\\mbox{diag}(a,a,b,0)$, while from section~\\ref{subsec_n-3} we know that {$\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and   $\\Phi_{ij}\\neq 0$ (Remark~\\ref{rem_6d}).} Substituting the form of $\\rhob$ into (\\ref{eqn:algI})--\\eqref{eqn:rhoC} {{and}} (\\ref{eqn:algII}) further reveals that $\\Phi_{ij}=\\diag(\\alpha,-\\alpha,0,0)\\neq 0$.\n\n\nFor $m=1$ one obviously has $\\rhob=\\mbox{diag}(a,0,0,0)\\neq 0$. From section~\\ref{subsec_m=1} we know that { $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and $\\Phi_{ij}\\neq 0$ (Remark~\\ref{rem_6d_m=1}).} \n\n\nFinally, for $m=2$ we can have two possibilities. If $\\Phi^A_{ij}\\neq0$ point \\eqref{RT} of theorem~\\ref{prop_GSHD} \nshows that $\\rhob=\\mbox{diag}(a,a,0,0)\\neq 0$. On the other hand, if $\\Phi^A_{ij}=0=\\Phi^S_{ij}$ we still get $\\rhob=\\mbox{diag}(a,a,0,0)\\neq 0$ from Proposition~\\ref{prop_IIabd}, whereas if $\\Phi^A_{ij}=0\\neq\\Phi^S_{ij}$ the more general form $\\rhob=\\mbox{diag}(a,b,0,0)$ is permitted (with $a\\neq 0\\neq b$).\n\n\nWe have checked in the various permitted cases  that the constraints (\\ref{eqn:algI})--\\eqref{eqn:rhoC} and \\eqref{alg_no_Phi4}--\\eqref{eqn:algII_N}\n allow for nontrivial type II Weyl tensor. {These results are summarized in table~\\ref{tab_6D}, together with a few examples. Examples of spacetimes with distinct non-zero eigenvalues follow from the general discussion of section~\\ref{subsubsec_examples_distinct}, which need not be repeated here.} \n\n\n\\begin{table}[t]\n  \\begin{center}\n  \\begin{tabular}{|c|l|c|c|c|l|c|}\n    \\hline Case & $m$ & Possible form of $\\rhob$ &  O.C. & O.S. & I.T.G. & Examples  \\\\\\hline\n    $\\Phi_{ij} \\not=0$ & 4 & $\\diag(a,a,a,a)$    & $\\surd$ & $\\surd *$ & ${\\cal D}_{23}*$, ${\\cal D}_{45}*$ & RT \\\\\n     $  $ & 3 & $\\diag(a,a,b,0)$     & if $b=a$ & X & ${\\cal D}_{23}$ (if $b\\neq a$) & ?($\\dag$) \\\\\n     $  $ & 2 \\ ($\\Phi^A_{ij}=0$) & $\\diag(a,b,0,0)$    & if $b=a$ &  & ${\\cal D}_{45}$ (${\\cal D}_{23}$ if $b=a$) & {dS$_4\\times$S$^2$} \\\\\n     $  $ & 2 \\ ($\\Phi^A_{ij}\\neq 0$) & $\\diag(a,a,0, 0)$    & $\\surd$ & $\\surd$ & ${\\cal D}_{23}$, ${\\cal D}_{45}$ & ? \\\\\n     $  $ & 1 & $\\diag(a,0,0,0)$    & $\\surd$ & X & & {dS$_3\\times$S$^3$} \\\\\n     $  $ & 0 & $\\diag(0,0,0,0)$   & $\\surd$ & &  & Kundt \\\\\\hline\n     $\\Phi_{ij} =0  $ & 4 & $\\diag(a,a,b,b)$    & if $b=a$ & $\\surd$ &  ${\\cal D}_{23}$, ${\\cal D}_{45}$ (if $b\\neq a$)& ?($\\dag$) \\\\\n     $    $ & 3 & X    & & & & X \\\\\n     $ $  & 2 & $\\diag(a,a,0,0)$  & $\\surd$ &  $\\surd$ & ${\\cal D}_{23}$, ${\\cal D}_{45}$ & ? \\\\\n     $ $  & 1 & X     & & & & X \\\\\n     $ $  & 0 & $\\diag(0,0,0,0)$  & $\\surd$ &  & & Kundt \\\\\n\\hline\n  \\end{tabular}\n    \\caption{Permitted forms of the optical matrix associated with a non-twisting mWAND in a six-dimensional type II spacetime in the two cases $\\Phi_{ij} \\not=0$  and $\\Phi_{ij}=0$ (subtype II(abd)). Recall that $\\Phi^A_{ij}\\neq 0$ is possible only for $m=2$. A question mark indicates subcases that are in principle permitted but for which no examples are known (so we do not claim that these classes are non-empty), while  X corresponds to forbidden subcases. \nAlthough we are not aware of examples with $m=3$ and $\\Phi_{ij} \\not=0(=\\Phi^A_{ij})$ having the most general structure, special solutions with $a=b$ (e.g., a static black string) can be obtained by warping once a type II 5d Einstein metric. {Similarly, generic $m=4$ solutions with $\\Phi_{ij}=0$ are not known, but in the special case $a=b$ they} exist within the \\RT family \\cite{PodOrt06} (the ``$\\mu=0$'' subcase). {These special examples are denoted by a $\\dag$ in the table.} In the O.C. column we have indicated if\/under what conditions $\\rhob$ satisfies the optical constraint. In the O.S. column whether ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{5},\\bl\\}$ defines an optical structure and in the I.T.G. column we listed integrable complex two-dimensional distribution that admit totally geodesic integral surfaces (cf. section~\\ref{subsec_integrability} and appendix~\\ref{app_shearfree}). The symbol $*$ means that the corresponding statement is true in a special subcase specified in the text (see section~\\ref{subsec_integrability}). \n\\label{tab_6D}}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Additional examples}\n\n\n\nIn addition to the examples mentioned in table~\\ref{tab_6D} and in section~\\ref{subsubsec_examples_distinct}, here we give  two more solutions with $m=1$ in six dimensions (and, necessarily, $\\Phi_{ij}\\neq0$). \n\n\\subsubsection{Example with $\\Lambda\\neq0$}\n\n\n\nWe can use, e.g., AdS$_4\\times$H$^2$ in the form \n\\be\n \\d s^2=\\frac{3\\beta^2}{z^2}(-\\d t^2+\\d\\rho^2+\\rho^2\\d\\phi^2+\\d z^2)+\\frac{\\beta^2}{x^2}(\\d x^2+\\d y^2) .\n\\ee \nFor the null congruence\n\\be\n \\ell_a\\d x^a=\\d t+\\d\\rho ,\n\\ee \nwith the spacelike frame vectors $\\bbm_{(2)}=z\\rho^{-1}\/(\\sqrt{3}\\beta)\\pa_\\phi$, $\\bbm_{(3)}=z\/(\\sqrt{3}\\beta)\\pa_z$, $\\bbm_{(4)}=x\\beta^{-1}\\pa_x$, $\\bbm_{(5)}=x\\beta^{-1}\\pa_y$,  one finds $\\rho_{ij}=\\mbox{diag}(a,0,0,0)$, where $a=z^2\/(3\\beta^2\\rho)$. \n\n\n\n\\subsubsection{Example with $\\Lambda$ arbitrary}\n\nOne can alternatively construct an example with {$\\rho_{ij}=\\mbox{diag}(a,0,0,0)$} by starting from a null geodesic congruence in dS$_3\\times$S$^2$ and adding an extra dimension by Brinkmann's warp. For example, \n\\be\n \\d s^2=f(z)\\left[-V(r)\\d t^2+\\frac{1}{V(r)}\\d r^2+r^2\\d\\chi^2+\\beta^2(\\d\\theta^2+\\sin^2\\theta\\d\\phi^2)\\right]+\\frac{\\d z^2}{f(z)}, \n\\ee \nwith the null congruence\n\\be\n \\ell_a\\d x^a=\\d t+V^{-1}\\d r ,\n\\ee \nwhere  \n\\be \nV(r)=1-\\frac{r^2}{2\\beta^2}, \\qquad f(z)=-\\lambda z^2+2dz+b , \\qquad d^2=\\frac{1}{4\\beta^2}-\\lambda b.\n\\ee\nUsing the spacelike frame vectors $\\bbm_{(2)}=f^{-1\/2}r^{-1}\\pa_\\chi$, $\\bbm_{(3)}=f^{-1\/2}\\beta^{-1}\\pa_\\theta$, $\\bbm_{(4)}=f^{-1\/2}\\beta^{-1}\\sin^{-1}\\theta\\pa_\\phi$, $\\bbm_{(5)}=f^{1\/2}\\pa_z$,  one finds $\\rho_{ij}=\\mbox{diag}(a,0,0,0)$, where $a=\\frac{1}{rf}$. \n\nIt follows from \\cite{PraPraOrt07,OrtPraPra11} that $\\Phi_{ij}\\neq0$, {as it should be}. In general $\\Lambda$ can have any sign but, in particular, we can set $\\Lambda=0$ by choosing the parameter $\\lambda=0$ in $f(z)$. \n\n\n\n\n\\subsection{Optical constraint, optical structures and totally geodesic complex two-dimensional null surfaces}\n\n\\label{subsec_integrability}\n\n\nFirst, in the non-twisting case the optical constraint \\eqref{canformL} is satisfied iff the optical matrix $\\rhob$ possesses only one (possibly repeated) non-zero eigenvalue plus, possibly, some vanishing eigenvalues. This explains column O.C. (``optical constraint'') of table~\\ref{tab_6D}.\n\nNext, when $\\bl$ is non-twisting one can always take an eigenframe of $\\rhob$ to be parallelly transported \\cite{OrtPraPra10,PraPra08} and thus in particular set $\\M{i}{j}{0}=0$. It is then obvious that to each pair of repeated eigenvalues of $\\rhob$ there corresponds an integrable two-dimensional totally null distribution, defined by two unit vectors of the corresponding eigenspace and by $\\bl$ (cf. eq.~\\eqref{D23_6d}). \n\nFurther conditions on the Ricci rotation coefficients can be obtained by considering the Ricci identity (11k,\\cite{OrtPraPra07}), i.e., eq.~\\eqref{11k}, similarly as in section~\\ref{sec_totgeod}.\nConsequences of this equation enable one to prove the statements in columns  O.S. (``optical structure'') and I.T.G. (``integrable totally geodesic'' complex two-dimensional subspace) of table~\\ref{tab_6D}. Since this is rather straightforward, let us just exemplify it to show that when $\\rho_{ij}=\\diag(a,a,b,b)$ (with $b\\neq a$) the maximally totally  null distribution  ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{5},\\bl\\}$ defines an optical structure (cf. eqs.~\\eqref{6D_OS}). Namely, from \\eqref{11k} with $i,j,k=224$ and $i,j,k=334$ one gets $\\M{4}{2}{2}=\\M{4}{3}{3}$, while $i,j,k=225$ and $i,j,k=335$ give $\\M{5}{2}{2}=\\M{5}{3}{3}$. Similarly, by taking $i,j,k=442\/552$ and $i,j,k=443\/553$ one obtains, respectively, $\\M{2}{4}{4}=\\M{2}{5}{5}$ and $\\M{3}{4}{4}=\\M{3}{5}{5}$. Next, with $i,j,k=324,234,325,235$ one finds $\\M{3}{4}{2}=\\M{2}{4}{3}=\\M{3}{5}{2}=\\M{2}{5}{3}=0$, while $i,j,k=524,425,534,435$ give $\\M{5}{2}{4}=\\M{4}{2}{5}=\\M{5}{3}{4}=\\M{4}{3}{5}=0$. This implies that eqs.~\\eqref{6D_OS} are satisfied, as we wanted to prove. \n\nLet us finally remark that in the Robinson-Trautman case, corresponding to the first row of table~\\ref{tab_6D}, eq.~\\eqref{11k} reduces to $L_{1i}=0$  \n(cf. also the results of \\cite{PodOrt06,PraPra08}) and thus does not constraint the $\\M{i}{j}{k}$. However, in the special subcase (cf.~\\cite{PodOrt06}) in which the transverse space is of constant curvature one can show by inspection that there exists a natural parallelly transported frame such that ${\\cal D}$ defines an optical structure and ${\\cal D}_{23}$, ${\\cal D}_{45}$ correspond to totally geodesic null two-surfaces.\\footnote{This is a straightforward extension of the discussion in the 5d case, see footnote~10 of \\cite{Ortaggioetal12}.} {{In fact the real $\\mbb{i}$ can be paired arbitrarily in this case, giving rise also to other optical structures and totally geodesic null two-surfaces, and since }} there is a  continuous freedom in  spins there exist {{actually}} infinitely many optical structures. \nThis partly answers a question raised in remark~5.3 of \\cite{Taghavi-Chabert11b}, by showing that also in even dimensions a spacetime may admit more than $2^{[n\/2]}$ optical structures (noting also that in this example replacing $\\bl\\leftrightarrow\\bn$ gives other optical structures). \n\n\n\n\n\\subsection{Type III\/N spacetimes}\n\nBy the results of \\cite{Pravdaetal04}, in this case there are only two admitted structures of $\\rhob$: either $\\rhob=0$ (Kundt spacetimes) or $\\rhob=\\diag(a,a,0,0)\\neq0$. Since eq.~\\eqref{11k} does not contain any Weyl components of zero boost weight, the above discussion still applies and implies that in the case $\\rhob=\\diag(a,a,0,0)\\neq0$ both ${\\cal D}_{23}$ and ${\\cal D}_{45}$ correspond to totally geodesic null two-surfaces and that ${\\cal D}$ defines an optical structure -- explicit examples can be constructed as explained in \\cite{OrtPraPra10}.\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to Lode Wylleman for useful discussions. The authors acknowledge support from research plan {RVO: 67985840} and research grant no P203\/10\/0749. \n\n\n\n\\renewcommand{\\thesection}{\\Alph{section}}\n\\setcounter{section}{0}\n\n\\renewcommand{\\theequation}{{\\thesection}\\arabic{equation}}\n\n\n\n\\section{Algebraic constraints for type II spacetimes}\n\\setcounter{equation}{0}\n\n\\label{app_alg_eq}\n\n\\subsection{{General mWAND}}\n\nFor completeness and future reference, let us give here some additional algebraic constraints following from the Bianchi and Ricci identities for type II Einstein spacetimes in arbitrary dimensions. \n\nNote that rhs of eq. \\eqref{A4} is not symmetric under $ij\\leftrightarrow kl$ and thus \\eqref{A4}$_{ijkl}-$\\eqref{A4}$_{klij}$ leads to an  algebraic equation\n\\be\n2(\\Phia_{ij} A_{kl}-\\Phia_{kl} A_{ij}) +\\Phi_{k[j}\\rho_{i]l}-\\Phi_{l[j}\\rho_{i]k}-\\Phi_{j[k}\\rho_{l]i}+\\Phi_{i[k}\\rho_{l]j}\n+\\Phi_{ij[k|s}\\rho_{s|l]}-\\Phi_{kl[i|s}\\rho_{s|j]}=0 , \\label{alg_no_Phi4}\n\\ee\nand its trace gives the antisymmetric part of \\eqref{B8}.\nBy multiplying by $\\rho_{lj}$ and using \\eqref{B8}--\\eqref{B8asym} and symmetries of the Weyl tensor, after  lengthy calculations one arrives at an equation with no $\\Phi_{ijkl}$ terms\n\\be\n\\rho \\Phi_{j[i|}\\rho_{j|k]}+\\Phi_{jl}\\rho_{jl}A_{ki}+\\rho_{jl}\\rho_{jl}\\Phia_{ik}\n+\\Phi_{jl}\\left[ 3\\rho_{[j|i}\\rho_{|l]k} +\\rho_{[k|l}\\rho_{j|i]} \\right]\n+2\\rho_{jl}\\left[ \\Phi_{j[i|}A_{l|k]}+2\\Phia_{l[i|}\\rho_{j|k]}\\right]=0.\n \\label{constr_twist}\n\\ee\n\n\\subsection{{Non-twisting mWAND with $\\Phia_{ij}=0$}}\n\nAs already mentioned, by differentiating algebraic constraints and using the Bianchi and Ricci identities one can arrive at further algebraic constraints.\nFor example, in the {\\em non-twisting} case {with $\\Phia_{ij}=0$},\\footnote{{{Recall that in the non-twisting case with $\\Phia_{ij}\\neq0$ the form of $\\rhob$ is fully determined by point \\eqref{PhiA} of theorem~\\ref{prop_GSHD}, cf.~\\cite{Ortaggioetal12}.}}}\nby differentiating~\\eqref{eqn:algI} \nwe arrive at \\eqref{eqn:rhoC} {(equivalent to setting $A_{ij}=0=\\Phia_{ij}$ in \\eqref{thornB8}), which can be rewritten as}\n\\be\n - [\\rhob^2] \\Phi_{ik} + \\Phi (\\rhob^2)_{ik} +  2 \\Phi_{ij} (\\rhob^2)_{jk} + {\\Phi}_{ijkl} {(\\rhob^2)}_{jl}  =0 \\qquad (A_{ij}=0{=\\Phia_{ij}}),\n \\label{eqn:algII_0}\n\\ee\nwhere $[\\rhob^2]$ denotes the trace of {$\\rhob^2$}, etc. The trace of \\eqref{eqn:algII_0} vanishes and multiplying \\eqref{eqn:algII_0} by $\\rho_{ik}$ gives\n\\be\n\t[\\rhob^2][\\Phib \\cdot \\rhob]-[\\rhob][\\Phib \\cdot\\rhob^2]=0 \\qquad {(A_{ij}=0{=\\Phia_{ij}})} .\n\t\\label{trace}\n\\ee\nDifferentiation {of \\eqref{eqn:algII_0}} leads to \n\\bea\n & & -3 \\left[ - [\\rhob^3] \\Phi_{ik} + \\Phi (\\rhob^3)_{ik} +  2 \\Phi_{ij} (\\rhob^3)_{jk} + {\\Phi}_{ijkl} {(\\rhob^3)}_{jl}   \\right] \\nonumber\\\\\n\t\t& & {}+ \\left( \\Phi [\\rhob^2] + [\\Phib \\cdot\\rhob^2]\n\t\t\\right) \\rho_{ik} \n- \\left(\\Phi \\rho + [\\Phib \\cdot \\rhob]\n\\right) (\\rhob^2)_{ik}=0 \\qquad (A_{ij}=0{=\\Phia_{ij}})\n\\label{eqn:algII} ,\n\\eea\nwith trace {of \\eqref{eqn:algII}} vanishing  due to \\eqref{trace}. \n\nFor $\\Phi_{ij}=0$, the $(N-1)$th derivative of  \\eqref{B8}   reads\n\\be\n\t{\\Phi}_{ijkl} {(\\rhob^N)}_{jl} =0, \\quad N =1,2,\\dots \\qquad (A_{ij}=0{=\\Phi_{ij}}).\n\t\\label{eqn:algII_N}\n\\ee\n\n\\section{{Non-twisting type III Einstein spacetimes in arbitrary dimension}}\n\n\n\\setcounter{equation}{0}\n\n\\label{app_type_N_III}\n\n\n\nFor non-twisting Einstein spacetimes of type N, $\\rho_{ij}=\\diag({a},{a},0,\\dots,0)$ \\cite{Pravdaetal04,Durkeeetal10}, where ${a}$ {vanishes} in the case of Kundt spacetimes. Let us present  here a new derivation of the form of the optical matrix for a non-twisting type III Einstein spacetime in arbitrary\ndimension. This is considerably simpler than the original derivation of \\cite{Pravdaetal04}. \n\nFor non-twisting type III, eq. (59) of \\cite{Pravdaetal04} reduces to {(in the notation of \\cite{Durkeeetal10})}\n\\be\n2 \\Psi'_{lij} \\rho_{lk} = \\rho \\Psi'_{kij}.   \\label{typeIIInon-twist}\n\\ee\nIn the Kundt case ($\\rhob=0$) this becomes an identity, while for non-twisting non-Kundt Einstein spacetimes (to which we restrict from now on) necessarily $\\rho\\neq0$ (cf. Proposition~1 of \\cite{OrtPraPra07}) and \\eqref{typeIIInon-twist} gives non-trivial information.\nFollowing \\cite{OrtPraPra10} we choose a parallelly propagated frame with diagonal form of  $\\rho_{ij}=\\diag(\\rho_2,\\rho_3,\\dots)$.\nWithout loss of generality we can assume that $\\Psi'_{2ij} \\not=0$ for some values of $i,\\ j$. Eq.~\\eqref{typeIIInon-twist} then implies\n\\be\n\\rho_2 = \\frac{\\rho}{2}{\\neq0} \\label{typeIIIrho}.\n\\ee\nUsing the $r$-dependence of the non-vanishing eigenvalues of $\\rho_{ij}$ (see eq.~\\eqref{rho_i}), this gives\n\\be\n\\frac{2}{r-b_2} = \\sum_p \\frac{1}{r-b_p},\n\\ee\nwhere $p$ corresponds to all non-vanishing eigenvalues of $\\rho_{ij}$.\nApart from the Kundt case with $\\rho_{ij}=0$, this equation can be satisfied only for $\\rho_{ij}$ of rank 2, with ${a\\equiv}\\rho_2=\\rho_3$.\nThus {\\em the optical matrix for the non-twisting type III Einstein spacetime is either vanishing (Kundt) or \n$\\rho_{ij}=\\diag({a},{a},0,\\dots,0)$}, exactly as in the type N case.\n\n\n\n\n\n\n\\section{On the geodetic and shearfree condition in four and higher dimensions}\n\n\\setcounter{equation}{0}\n\n\\label{app_shearfree}\n\n\n\\subsection{Goldberg-Sachs in four dimensions}\n\n\\label{app_GS_4D}\n\n\nThe four-dimensional Goldberg-Sachs theorem is usually expressed as a statement about a null congruence being geodesic ($\\kappa=0$) and shearfree ($\\sigma=0$). As well-known, this property can be interpreted as different but equivalent geometrical statements, which we now briefly review. This will later be useful for discussing the higher dimensional case.\n\n\n\n\\subsubsection{Integrability of two-spaces}\n\n\\label{subsubsec_integr_d4}\n\nBy considering the commutator $[\\delta,D]$ (see, e.g., p.~77 of \\cite{Stephanibook})\none immediately sees that  $\\bl$ is geodesic and shearfree if and only if the distribution\n\\be\n {\\cal D}=\\mbox{Span}\\{\\bbm,\\bl\\} \n\\ee\nis integrable (cf.~\\cite{penrosebook2}), i.e. $[{\\cal D},{\\cal D}]\\subset{\\cal D}$.\nA similar conclusion holds of course for $\\bar{\\cal D}$. The geodesic{\\&}shearfree property can thus be seen as a statement about the null bivector $\\bl\\wedge\\bbm$ instead of one about the vector field $\\bl$. While the vector field $\\bl$ defines a privileged real null direction lying in ${\\cal D}\\cap\\bar{\\cal D}$, $\\bbm$ is not fixed uniquely (null rotations with $\\bl$ fixed, spins and boost can be performed, under which $\\bl\\wedge\\bbm$ is fixed, up to a rescaling, and $\\kappa=0=\\sigma$ is preserved). The significance of this viewpoint appears clearly when considering complex extensions of the Goldberg-Sachs theorem \\cite{PlebHac75}.\n\n\n\\subsubsection{Conformal structure of the screen space}\n\nGiven a generic null vector field $\\bl$, its flow preserves the conformal structure of the ``screen space'' $L^\\bot\/L$ when \\cite{RobTra83} (see also \\cite{NurTra02,Trautman02a,Trautman02b})\n\\be \n {\\cal L}_\\l g=\\rho g+\\l\\otimes\\xi+\\xi\\otimes\\l  \\quad \\Leftrightarrow  \\quad \\kappa=0=\\sigma .\n \\label{Lie_4}\n\\ee\n\nIn four dimensions only, this is equivalent to requiring that the complex structure of the screen space is preserved by the flow of $\\bl$ \\cite{RobTra83,NurTra02,Trautman02a,MasTag08}.\n\nThis condition can also be derived by imposing Maxwell equations on a null field (two-form) \\cite{robinsonnull} (cf. also, e.g., \\cite{Stephanibook,penrosebook2,NurTra02,Trautman02a,Trautman02b}).\n\n\n\n\\subsection{Conformal structure of the screen space: the standard geodesic{\\&}shearfree condition}\n\n\\label{subsec_screenHD}\n\nRef.~\\cite{RobTra83} (see also \\cite{Trautman02b}) proposed to extend (\\ref{Lie_4}) to higher dimensions as a definition of the geodesic{\\&}shearfree condition. As it turns out,\n\\be \n {\\cal L}_\\l g=\\rho g+\\l\\otimes\\xi+\\xi\\otimes\\l  \\quad \\Leftrightarrow  \\quad \\kappa_{i}=0=S_{ij}-\\frac{S_{kk}}{n-2}\\delta_{ij} ,\n \\label{Lie}\n\\ee\nso that indeed $\\bl$ is geodesic and shearfree in the standard sense. Some comments on algebraically special Einstein spacetimes admitting such a congruence have been already given in section~\\ref{subsec_OC} and appendix~\\ref{app_shearfreetwist}.\n\n\n\nNote that for $n>4$ eq.~(\\ref{Lie}) does {\\em not} follow from Maxwell's equations for a null field (intended again as a two-form), while geodeticity {($\\kappa_{i}=0$)} does, and shear is non-zero for expanding solutions \\cite{Ortaggio07,Durkeeetal10}.\n\n\n\n\n\\subsection{Existence of an optical structure}\n\n\\label{subsubsec_optical}\n\n\n\\subsubsection{Complex notation}\n\n\n\nFor later discussions it will be useful to use a complex basis. Namely, in {\\em even} dimensions one can take the frame $(\\bl,\\bn,\\bmu_A,\\bar\\bmu_A)$ and in {\\em odd} dimensions the frame $(\\bl,\\bn,\\bmu_A,\\bar\\bmu_A,\\bx)$. Except for the unit spacelike vector $\\bx$, all the vectors are null, the complex $\\bmu_A$ are defined by\n\\be\n  \\bmu_2=\\frac{1}{\\sqrt{2}}(\\mbb{2}+i\\mbb{3}) , \\qquad \\bmu_4=\\frac{1}{\\sqrt{2}}(\\mbb{4}+i\\mbb{5}) , \\qquad \\ldots ,\n  \\label{mu}\n\\ee\nand ${\\bar\\bmu_A}$ by their complex conjugates, where $A=2\\mu$, $\\mu=1,\\ldots,(n-2-\\epsilon)\/2$, with $\\epsilon=0,1$ for even and odd dimensions, respectively (in 4d this would be the standard NP frame). The metric becomes\n\\be\n g=\\bl\\otimes \\bn+\\bn\\otimes\\bl+\\bmu_A\\otimes\\bar\\bmu_A+\\bar\\bmu_A\\otimes\\bmu_A+\\epsilon \\bx\\otimes\\bx .\n \\label{complex_g}\n\\ee\n\nThen one can define the complex counterpart of the Ricci rotation coefficients. These are defined as an obvious extension of the real coefficients, e.g.,\n\\be\n \\cL_{AB}=\\mu^a_A\\mu^b_B\\nabla_b l_{a} , \\qquad \\cL_{A\\bar B}=\\mu^a_A\\bar\\mu^b_B \\nabla_b l_{a} , \\qquad \\cM{A}{C}{B}=\\mu^a_C\\mu^b_{{B}}\\nabla_b \\mu_{A a}, \n\\ee\nand other coefficients (and their complex-conjugates) are defined similarly.\n\n\n\n\\subsubsection{Optical structure} \n\n\n\\label{subsubsec_OS}\n\nRef.~\\cite{HugMas88} studied the consequences of Maxwell's equations for a null field defined as an $n\/2$-form in $n$ even dimensions. From these, they arrived at a generalization of the geodesic{\\&}shearfree condition different from the $\\kappa=0=\\sigma$ condition discussed in section~\\ref{subsec_screenHD} (see also \\cite{NurTra02,Trautman02a,Trautman02b,MasTag08}). Recently, this has been extended also to odd dimensions \\cite{Taghavi-Chabert11}. Namely, consider the totally null  $(n-\\epsilon)\/2$-dimensional distribution (recall that $A$ in $\\mu_A$ can be only even)\n\\be\n {\\cal D}=\\mbox{Span}\\{\\bl,\\bmu_2,\\ldots,\\bmu_{(n-2-\\epsilon)}\\} .\n \\label{D_gen}\n\\ee\nIf ${\\cal D}$ and ${\\cal D}^\\bot$ are integrable, i.e.,\n\\be\n [{\\cal D},{\\cal D}]\\subset{\\cal D} , \\qquad [{\\cal D}^\\bot,{\\cal D}^\\bot]\\subset{\\cal D}^\\bot ,\n \\label{Robinson}\n\\ee\n${\\cal D}$ is said to define an ``{\\em optical structure}'' \\cite{Taghavi-Chabert11} (note that ${\\cal D}^\\bot={\\cal D}$ in even dimensions).  In 4d, eq.~\\eqref{Robinson} reduces to the standard conditions $\\kappa=0=\\sigma$ (as discussed in section~\\ref{subsubsec_integr_d4}), which indeed corresponds to the conditions coming from the Mariot-Robinson theorem \\cite{Stephanibook,penrosebook2} for a Maxwell null two-form. In  higher dimensions, using the complex counterpart of the commutators given in \\cite{Coleyetal04vsi}, one finds that (\\ref{Robinson}) is equivalent to \n\\be\n \\cL_{A0}=\\cL_{x0}=\\cL_{AB}=\\cL_{Ax}=\\cL_{xA}=\\cM{A}{B}{0}=\\cM{A}{x}{0}=\\cM{A}{B}{C}=\\cM{A}{B}{x}=\\cM{x}{A}{B}=0 .\n \\label{optical_complex}\n\\ee\n(In fact one first obtains conditions such as $\\cM{A}{B}{C}=\\cM{C}{B}{A}$, but $\\cM{A}{B}{C}=-\\cM{B}{A}{C}$, etc.) The equations above represent the conditions obtained when $n$ is odd, however, if $n$ is even one can simply drop all equations containing $x$ (this will be understood from now on).\n\nNote, in particular, that the vector $\\bl$ is {\\em geodesic} ($\\cL_{A0}=\\cL_{x0}=0$), but generally is not required to be shearfree (except when $n=4$). \nIn even dimensions it has also been observed that (\\ref{Robinson}) means that the complex structure of the screen space is preserved \\cite{NurTra02,Trautman02a,MasTag08}. We have shown in \\cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes possesses an optical structure. In appendix \\ref{app_OS5d} we extend the results of \\cite{Ortaggioetal12} by showing that, in fact, all algebraically special 5d Einstein spacetimes possess (at least) one optical structure.\n\n\n\n\\paragraph{Optical structure in six dimension} Let us rewrite the conditions \\eqref{optical_complex} in real notation in the special case $n=6$, which is useful for the discussion in the main text (see \\cite{Ortaggioetal12} for the case $n=5$). One readily gets\n\\beqn\n & & \\kappa_{i}=0 \\quad (i=2,\\ldots,5) , \\nonumber \\\\\n & & \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\quad \\rho_{44}=\\rho_{55}, \\quad \\rho_{45}=-\\rho_{54},  \\nonumber \\\\\n & & \\rho_{24}=\\rho_{35}, \\quad \\rho_{42}=\\rho_{53}, \\quad \\rho_{34}=-\\rho_{25}, \\quad \\rho_{43}=-\\rho_{52}, \\nonumber \\\\\n & & \\M{2}{4}{0}=\\M{3}{5}{0}, \\quad \\M{3}{4}{0}=-\\M{2}{5}{0}, \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad (n=6) \\label{6D_OS} \\\\\n & & \\M{2}{4}{2}-\\M{3}{4}{3}=\\M{2}{5}{3}+\\M{3}{5}{2}, \\quad -\\M{2}{5}{2}+\\M{3}{5}{3}=\\M{2}{4}{3}+\\M{3}{4}{2}, \\nonumber \\\\ \n & & \\M{2}{4}{4}-\\M{2}{5}{5}=\\M{3}{4}{5}+\\M{3}{5}{4}, \\quad \\M{2}{4}{5}+\\M{2}{5}{4}=-\\M{3}{4}{4}+\\M{3}{5}{5} . \\nonumber \n\\eeqn\n\nIn particular, if $\\bl$ is twistfree then $\\rhob$ has two pairs of repeated eigenvalues.\n\n\n\n\n\n\\subsection{Optical constraint}\n\n\n\nBased on results for Kerr-Schild spacetimes, Ref.~\\cite{OrtPraPra09} put forward yet another possible generalization of the shearfree condition for mWANDs in higher dimensions. This is the so called ``optical constraint'', already discussed in section~\\ref{subsec_OC}, which involves only the null direction $\\bl$ (as opposed to the optical structure discussed above). The Lorentz transformation freedom of null rotations preserving $\\bl$, boosts and spins is thus retained in this case (indeed spins can be used to arrive at the canonical form~\\eqref{canformL}), see also \\cite{OrtPraPra10} for related comments. We have shown in \\cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes admits an mWAND obeying the optical constraint, {{and Ref.~\\cite{Wylleman_priv} extended our result to prove that in fact {\\em all} algebraically special Einstein spacetimes admit such an mWAND} (see also \\cite{OrtPraPra12rev})}.\nNote that in 4d the optical constraint is a necessary condition for $\\bl$ to be a repeated principal null direction but is not {sufficient} \\cite{Ortaggioetal12}.  \n  \n\n\\subsection{Integrability of a (complex) two-dimensional totally null distribution}  \n\n\\subsubsection{${\\cal D}_{23}$ integrable}\n\nAs a further generalization of the geodesic{\\&}shearfree condition one can consider the integrability of the complex two-dimensional totally null distribution\n\\be \n {\\cal D}_{23}=\\mbox{Span}\\{\\bmu_{2},\\bl\\} .\n \\label{D}\n\\ee\nIt is easy to show that ${\\cal D}_{23}$ is integrable if and only if \n\\be\n \\cL_{20}=0, \\qquad \\cL_{B2}=\\cM{2}{B}{0}, \\qquad \\cL_{\\bar B2}=\\cM{2}{\\bar B}{0} \\quad (B\\neq 2) , \\qquad \\cL_{x2}=\\cM{2}{x}{0} .\n \\label{2integr}\n\\ee\nIn 4d this again reduces to the standard $\\kappa=0=\\sigma$ condition.\n\nIn particular, in six dimensions \\eqref{2integr} can be rewritten in real notation as\n\\beqn\n & & \\kappa_{2}=0=\\kappa_{3} , \\qquad  \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\nonumber \\label{D23_6d} \\\\\n & & \\rho_{42}=\\M{2}{4}{0}, \\quad \\rho_{43}=\\M{3}{4}{0}, \\quad \\rho_{52}=\\M{2}{5}{0}, \\quad \\rho_{53}=\\M{3}{5}{0}  \\qquad\\qquad (n=6) . \n\\eeqn\n\n\\subsubsection{${\\cal D}_{23}$ integrable with totally geodesic integral surfaces}  \n\nWe can strengthen the above conditions by further requiring the integral surfaces of ${\\cal D}_{23}$ to be {\\em totally geodesic}. The corresponding equations read\n\\beqn\n & & \\cL_{B0}=0=\\cL_{x0}, \\qquad \\cL_{B2}=0=\\cL_{x2}, \\qquad \\cL_{\\bar B2}=0 \\quad (B\\neq 2) , \\nonumber \\\\\n & & \\cM{2}{B}{0}=0=\\cM{2}{x}{0}, \\qquad \\cM{2}{B}{2}=0=\\cM{2}{x}{2}, \\qquad \\cM{2}{\\bar B}{0}=0 \\quad (B\\neq 2) , \\qquad \\cM{2}{\\bar B}{2}=0 \\quad (B\\neq 2) .\n \\label{2totgeod}\n\\eeqn\nNow $\\bl$ is necessarily geodesic ($\\cL_{B0}=0=\\cL_{x0}$). In 4d \\eqref{2totgeod} is equivalent to \\eqref{2integr} because $B=2$ is the only possibility (and there are no $x$-components). In real notation \\eqref{2totgeod} can be rewritten as (where we define $\\hat k\\neq 2,3$)\n\\beqn\n & & \\kappa_{i}=0 , \\qquad  \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\qquad \\rho_{\\hat k2}=0=\\rho_{\\hat k3} \\nonumber \\\\\n & & \\M{2}{\\hat k}{0}=0=\\M{3}{\\hat k}{0} , \\label{2totgeod_real} \\\\\n & & \\M{2}{\\hat k}{2}=\\M{3}{\\hat k}{3}, \\quad  \\M{2}{\\hat k}{3}=-\\M{3}{\\hat k}{2} \\nonumber .\n\\eeqn\n\n\n\n\n\n  \n\n\\section{Optical structures in five dimensions}\n\n\\label{app_OS5d}\n\n\nProposition~4 of \\cite{Ortaggioetal12} gives a set of sufficient conditions for a five-dimensional Einstein spacetime of type II or more special to possess an {\\em optical structure} \\cite{Taghavi-Chabert11}. In particular, it shows that, except possibly for Kundt spacetimes (and for a special subclass ``(iii)'' of genuine type II, later proven not to exist \\cite{Wylleman_priv}), all algebraically special Einstein spacetimes admit an optical structure in five dimensions. It is the purpose of this  appendix to show that this in fact holds also for Kundt spacetimes. (Obviously, if a complex optical structure is integrable its complex conjugate is integrable too and this will be understood in the following.) Combining this with the proof of \\cite{Wylleman_priv} that the special subclass (iii) of genuine type II is empty, we arrive at \n\\begin{prop}\n\\label{prop_integrab_5D}\nIn a five-dimensional Einstein spacetime admitting a multiple WAND~$\\lb$ there always exists an optical structure. In the case of type D spacetimes there exist in fact (at least) two optical structures. \n\\end{prop}\n\n\n\\begin{proof}\nLet us  show that Einstein spacetimes of the Kundt class always possess an optical structure in five dimensions. In other words, we need to show that in such spacetimes there always exists a null frame $\\{\\bl,\\bn,\\mbb{2},\\mbb{3},\\mbb{4}\\}$ such that \nthe totally null distribution\n\\be\n {\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\bl\\} ,\n\\ee\nand its orthogonal complement \n\\be\n {\\cal D}^\\bot=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4},\\bl\\} ,\n\\ee \nare both integrable. This is equivalent to \\cite{Ortaggioetal12} (see also appendix~\\ref{app_shearfree}, and \n{\\eqref{L1i_M}} for the definition of $\\M{a}{b}{c}$)\n\\beqn\n & & \\kappa_{i}=0, \\qquad \\rho_{33}=\\rho_{22}, \\qquad \\rho_{32}=-\\rho_{23},\\qquad \\rho_{24}=0=\\rho_{34}, \\qquad \\rho_{42}=0=\\rho_{43},  \t\t\t  \\label{Dorth_integr_k_rho} \n \\\\  & & \\M{2}{4}{0}=0=\\M{3}{4}{0} , \\qquad \\M{2}{4}{2}=\\M{3}{4}{3}, \\qquad \\M{2}{4}{3}=-\\M{3}{4}{2} .\n \\label{Dorth_integr_M} \n\\eeqn\n\n\nKundt spacetimes admit a metric in the form \\cite{Coleyetal03,ColHerPel06,PodZof09} \n\\be\n \\d s^2 =2\\d u\\left[\\d r+H(u,r,x)\\d u+W_\\alpha(u,r,x)\\d x^\\alpha\\right]+ g_{\\alpha\\beta}(u,x) \\d x^\\alpha\\d x^\\beta , \\label{Kundt_gen}\n\\ee\nwhere $\\alpha,\\beta=2,3,4$ in five dimensions. Here $\\bl=\\partial_r$ is a geodesic, twistfree, shearfree, non-expanding mWAND, so that \\eqref{Dorth_integr_k_rho} is automatically satisfied. Now, define a null frame with $\\ell_a\\d x^a=\\d u$, $n_a\\d x^a=\\d r+H\\d u+W_\\alpha\\d x^\\alpha$ and the spacelike vectors $\\mbb{i} $ living in the three-dimensional transverse (Euclidean) space spanned by the $x^\\alpha$ (their components will be, in particular, independent of $r$). Then one immediately finds\n\\be\n \\M{i}{j}{0}=0, \n\\ee\nso that the first of \\eqref{Dorth_integr_M} is satisfied in this frame. \n\nNext, using the Christoffel symbols given in \\cite{PodZof09} one can check that  \n\\be\n m_{(i)\\a;\\beta}=m_{(i)\\a||\\beta} ,\n\\ee\nwhere the covariant derivatives on the r.h.s. is taken w.r.t. the transverse metric $g_{\\a\\beta}$. \n\nIt follows that\n\\be\n {\\M{i}{j}{k}} ={\\Mt{i}{j}{k}} , \n\\ee\nwhere the connection coefficients on the r.h.s. are those computed w.r.t. the transverse metric $g_{\\a\\beta}$ (and the o.n. frame vectors of the transverse space are simply the ``projections'' $\\tilde m_{(i)\\a}$  of the $m_{(i)\\a}$ vectors). \n\nOne then finds\n\\be\n [\\mbtt{2}+i\\mbtt{3},\\mbtt{4}]=(\\Mt{4}{2}{4}+i\\Mt{4}{3}{4})\\mbtt{4} +[-\\Mt{2}{4}{2}+i(\\Mt{2}{3}{4}-\\Mt{2}{4}{3})]\\mbtt{2}+[\\Mt{3}{2}{4}-\\Mt{3}{4}{2}-i\\Mt{3}{4}{3}]\\mbtt{3} ,\n\\ee\nso that $\\mbox{Span}\\{\\mbtt{2}+i\\mbtt{3},\\mbtt{4}\\}$ is integrable iff\n\\be\n \\Mt{3}{4}{3}=\\Mt{2}{4}{2} , \\qquad \\Mt{3}{4}{2}=-\\Mt{2}{4}{3} .\n \\label{integr_234}\n\\ee \nProving the integrability of ${\\cal D}$ and ${\\cal D}^\\bot$ is thus now reduced to proving the integrability of $\\mbox{Span}\\{\\mbtt{2}+i\\mbtt{3},\\mbtt{4}\\}$. The transverse frame is arbitrary, so we just need to show that there exists at least one frame satisfying this integrability property.\n\nIf we define the complex null vector field\n\\be\n \\bmu=\\frac{1}{\\sqrt{2}}(\\mbtt{2}+i\\mbtt{3}) , \n \\label{def_mu}\n\\ee\nusing \\eqref{integr_234} the required integrability condition $[\\bmu,\\mbtt{4}]=\\alpha\\bmu+\\beta\\mbtt{4}$ reads\n\\be\n \\mu_{a||b}m_{(4)}^a\\mu^b=0 .\n\\ee \nThus by choosing a complex null geodesics $\\bmu$ in the transverse three-space we automatically obtain  the integrability of the corresponding  distributions ${\\cal D}$ and ${\\cal D}^\\bot$ and {(together with the result of \\cite{Wylleman_priv})} our proof is complete.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Shearfree twisting spacetimes (even dimensions)}\n\n\\label{app_shearfreetwist} \n\nTwisting geodesic mWANDs with zero shear are forbidden in odd dimensions \\cite{OrtPraPra07} but they are permitted in {\\em even} dimensions and they have necessarily $\\det(\\rhob)\\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\\propto\\delta_{ij}$). Here we present an explicit example in six dimensions. {To our knowledge, this is the first such example that has been identified.} \n\nFirst, consider the six-dimensional Ricci flat Taub-NUT metric \\cite{ManSte04} \n\\beqn\n \\d s^2= & & -F(r)(\\d t-2n_1\\cos\\theta_1\\d\\phi_1-2n_2\\cos\\theta_2\\d\\phi_2)^2+\\frac{\\d r^2}{F(r)} \\nonumber \\\\\n & & {}+(r^2+n_1^2)(\\d\\theta_1^2+\\sin\\theta_1^2\\d\\phi_1^2)+(r^2+n_2^2)(\\d\\theta_2^2+\\sin\\theta_2^2\\d\\phi_2^2) ,\n \\label{MasSte6D}\n\\eeqn\nwhere\n\\be\n F(r)=\\frac{r^4\/3+(n_1^2+n_2^2)r^2-2mr-n_1^2n_2^2}{(r^2+n_1^2)(r^2+n_2^2)} .\n\\ee\nWe observe that this is a spacetime of type D. A geodetic mWAND is given by\n\\be\n \\ell_a\\d x^a=\\d t+F(r)^{-1}\\d r-2n_1\\cos\\theta_1\\d\\phi_1-2n_2\\cos\\theta_2\\d\\phi_2 ,\n\\ee \nwhile a second one can simply be obtained by reflecting $\\bl$ as $t\\to-t$, $\\phi_1\\to-\\phi_1$, $\\phi_2\\to-\\phi_2$ \\cite{PraPraOrt07}. Using the frame vectors\n\\beqn\n & & m_{(2)a}\\d x^a=\\sqrt{r^2+n_1^2}\\d\\theta_1, \\qquad m_{(3)a}\\d x^a=\\sqrt{r^2+n_1^2}\\sin\\theta_1\\d\\phi_1, \\nonumber \\\\\n & & m_{(4)a}\\d x^a=\\sqrt{r^2+n_2^2}\\d\\theta_2, \\qquad m_{(5)a}\\d x^a=\\sqrt{r^2+n_2^2}\\sin\\theta_2\\d\\phi_2,  \n\\eeqn\none finds\n\\be\n\\rhob = \\left(\n\\begin{array}{cccc} \n      \\displaystyle \\frac{r}{r^2+n_1^2} & \\displaystyle -\\frac{n_1}{r^2+n_1^2} & 0 & 0\\\\ \n\t\t\t\\displaystyle \\frac{n_1}{r^2+n_1^2} & \\displaystyle \\frac{r}{r^2+n_1^2}  & 0 & 0 \\\\ \n\t\t\t0 & 0 & \\displaystyle \\frac{r}{r^2+n_2^2} & \\displaystyle -\\frac{n_2}{r^2+n_2^2}\\\\ \n\t\t\t0 & 0 & \\displaystyle \\frac{n_2}{r^2+n_2^2} & \\displaystyle \\frac{r}{r^2+n_2^2} \n  \\end{array}\n \\right) . \n\\ee\n\n\nOne can easily check that $\\rhob$  obeys the optical constraint~(\\ref{OC2}). {Moreover, we also observe that in the spacetime~\\eqref{MasSte6D} the maximally totally null distribution ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{4},\\bl\\}$\ndefines an {\\em optical structure}  (concept introduced and discussed in \\cite{HugMas88,NurTra02,Trautman02a,Trautman02b,Taghavi-Chabert11,Taghavi-Chabert11b}), and both the totally null distributions ${\\cal D}_{23}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\bl\\}$ and ${\\cal D}_{45}=\\mbox{Span}\\{\\mbb{4}+i\\mbb{5},\\bl\\}$ are integrable, with totally geodesic integral surfaces (see appendix~\\ref{app_shearfree} for the corresponding definitions and conditions).}\nThe null vector field $\\bl$ is generically shearing ($S_{22}=S_{33}\\neq S_{44}=S_{55}$), however in the special case $n_1=n_2$ (corresponding to the solutions of \\cite{AwaCha02}) it becomes {\\em shearfree} (while still being expanding and twisting). The class of shearfree twisting spacetimes is thus non-empty in higher dimensions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{Non-degenerate, non-twisting geodesic mWANDs violating the optical constraint ($n\\ge 7$)}}\n\n\\setcounter{equation}{0}\n\n\\label{app_violating}\n\n\nIn the main text we have seen examples of Einstein spacetimes admitting a non-twisting mWAND violating the optical constraint, see e.g. table~\\ref{tab_distinct_eigenval}. However, all of them have a degenerate $\\rhob$ (i.e., $m<n-2$). In this appendix we provide also some examples with $m=n-2$, relevant to theorem~\\ref{prop_GSHD} (point \\eqref{Phi0}) and proposition~\\ref{prop_non_deg2} (point 2.).\n\n\n\\subsection{Metric}\n\nUsing the theory of conformal Einstein spaces \\cite{Brinkmann25} (reviewed, e.g., in \\cite{petrov}), by taking a double Brinkmann warp one can construct the following Einstein space  (satisfying $R_{ab}=2\\Lambda g_{ab}\/(n-2)$):\n\\be\n \\d s^2=\\lambda r^2\\d u^2+2\\d u\\d r+(\\lambda ur-1)^2\\d\\sigma_\\lambda^2+r^2\\d\\Sigma_0^2 , \n \\label{general_violating}\n\\ee\nwhere \n\\be\n \\lambda=2\\frac{\\Lambda}{(n-1)(n-2)} ,\n\\ee\n$\\d\\sigma_\\lambda^2$ is a Riemannian Einstein space of dimension $n_\\sigma$ with Ricci scalar $R_\\sigma=n_\\sigma(n_\\sigma-1)\\lambda$ and $\\d\\Sigma_0^2$ a Ricci-flat Riemannian space of dimension $n_\\Sigma$. In order for $\\d s^2$ to have a non-zero Weyl tensor, the metrics $\\d\\sigma_\\lambda^2$ or $\\d\\Sigma_0^2$ cannot be both of constant curvature, so that either $n_\\sigma\\ge 4$ or $n_\\Sigma\\ge 4$ (or both). The complete spacetime has thus dimension $n=2+n_\\sigma+n_\\Sigma\\ge 7$ (unless it is of constant curvature, which is of no interest to us).\n\nLet us define the vector field $\\bl$ with covariant and contravariant components\n\\be\n \\ell_a\\d x^a=\\d u , \\qquad \\ell^a\\pa_a=\\pa_ r ,\n \\label{l_ex}\n\\ee\nwhich is obviously  null and hypersurface-orthogonal, thus automatically geodesic and twistfree, and $r$ is an affine parameter along it. This can be accompanied by another null vector $\\bn$ (also hypersurface-orthogonal)\n\\be\n n_a\\d x^a=\\d r+\\frac{\\lambda}{2}r^2\\d u ,\n \\label{n_ex}\n\\ee\nwhich satisfies the normalization condition $\\bl\\cdot\\bn=1$.\n\n\n\n\\subsection{Weyl tensor}\n\nFrom the results of \\cite{Brinkmann25} together with those on direct product spacetimes \\cite{PraPraOrt07} it readily follows that\n\\be\n C_{uabc}=0=C_{rabc} .\n \\label{C_ur_ex}\n\\ee \n\nLet us consider a set of $n-2$ spacelike o.n. vectors $\\bbm_{(i)}$ orthogonal to $\\bl$ and $\\bn$, composed of a subset of vectors denoted by $i=A, B, \\ldots$   living in the subspace of $\\d\\sigma_\\lambda^2$ and a subset denoted by $i=I,J, \\ldots$ living in the subspace of $\\d\\Sigma_0^2$. Then the only non-zero Weyl frame components are given by $C_{ABCD}$ and $C_{IJKL}$, and their $r$-dependence is (recall the first definition in \\eqref{bw0}) \n\\be\n \\Phi_{ABCD}=\\frac{1}{(\\lambda ur-1)^2}C^0_{ABCD} , \\qquad \\Phi_{IJKL}=\\frac{1}{r^2}C^0_{IJKL} , \n \\label{C_ABIJ_ex}\n\\ee\nwhere quantities with superscript $^0$ do not depend on $r$ (these are in fact the Weyl components of the respective ``subspaces''). \n\nIn particular, one has\n\\be\n \\WD{ij} =0 .\n\\ee\nNote also that one has generically $W_{ij}\\neq 0$ (recall  definition \\eqref{def_W} and constraint~\\eqref{W_phi}).\n\n\n\n\n\\subsection{Multiple WAND(s) and optics}\n\nFrom the above results it follows that both $\\bl$ and $\\bn$ are {double} WANDs and the spacetime is thus of type~D. In fact a bit more than that, since \n\\be\nC_{abcd}\\ell^d=0=C_{abcd}n^d \\label{Wandcond}\n\\ee\n (cf. \\cite{PraPra05,Ortaggio09} for the meaning of such conditions in terms of the Weyl type)  and the type is D(abd).\\footnote{The construction {of metric \\eqref{general_violating}} is carried out more naturally in a different coordinate system (not adapted to $\\bl$) in which $\\bl$ and $\\bn$ are related by a coordinate transformation (``time-reflection'') that leaves the metric invariant: therefore they share the same geometric properties.}\n\n\nThe optical matrix $\\rhob$ of $\\bl$ is obviously symmetric, it is moreover diagonal and non-degenerate with components\n\\be\n \\rho_{AB}=\\frac{\\lambda u}{\\lambda ur-1}\\delta_{AB} , \\qquad \\rho_{IJ}=\\frac{1}{r}\\delta_{IJ} .\n\\ee\nThere are thus two eigenvalue-blocks of dimension $n_\\sigma$ and $n_\\Sigma$, or $[n_\\sigma,n_\\Sigma]$. For instance, for $n=7$ we can construct explicit solutions with eigenvalues $\\{a,a,a,a,b\\}$, for $n=8$ we can have \n$\\{a,a,a,a,a,b\\}$ and $\\{a,a,a,a,b,b\\}$ and so on (in all cases $a\\neq0\\neq b$, $a\\neq b$). These provide examples for of the shearing spacetimes of  theorem~\\ref{prop_GSHD}  \nand proposition~\\ref{prop_non_deg2}, although an eigenvalue structure {more general than the one of metric \\eqref{general_violating} can in principle exist}. {In particular, the optical constraint is clearly violated (cf. eq.~\\eqref{canformL}).} Note also that we cannot construct a six-dimensional example with this method. By setting the cosmological constant to zero in the above metrics ($\\lambda=0$) one obtains Ricci-flat spacetimes -- however, these are  direct products and $\\rhob$ becomes degenerate, which is of no interest to the present discussion.\n\nBy taking direct products one can trivially generate many similar examples with a degenerate $\\bl$. These will have $\\WDS{ij} \\neq 0$, however still with $\\WDA{ij} =0$ (see \\cite{PraPraOrt07}), in agreement with point (i) of theorem~\\ref{prop_GSHD}. \n\n\n\\subsubsection{On possible additional (m)WANDs}\n\n\\label{subsubsec_additionalW}\n\n\n\nOne might wonder whether spacetimes \\eqref{general_violating} admit other multiple (double) WANDs (different from $\\bl$ and $\\bn$, eqs.~\\eqref{l_ex} and \\eqref{n_ex}) and in particular whether those can obey the optical constraint. Take a generic null vector $\\bk=\\alpha\\pa_u+\\beta\\pa_r+\\gamma^A\\bbm_{(A)}+\\gamma^I\\bbm_{(I)}$ (where the $\\bbm_{(A)}$ [$\\bbm_{(I)}$] have components only in the subspace of $\\d\\sigma_\\lambda^2$ [$\\d\\Sigma_0^2$]), i.e., $\\alpha^2\\lambda r^2+2\\alpha\\beta+\\delta_{AB}\\gamma^A\\gamma^B+\\delta_{IJ}\\gamma^I\\gamma^J=0$. Since for $\\gamma^A=0=\\gamma^I$ $\\bk$ gives the directions of $\\bl$ and $\\bn$ (respectively for $\\alpha=0$ and $\\beta=-\\frac{1}{2}\\alpha\\lambda r^2$), we now assume that $\\gamma^A$ and $\\gamma^I$ are not both zero (so that $\\alpha\\neq0$). Using \\eqref{C_ur_ex}, \\eqref{C_ABIJ_ex} and the Bel-Debever criteria \\cite{Ortaggio09} it follows that: (i) $\\bk$ is a WAND $\\Leftrightarrow C^0_{ABCD}\\gamma^A\\gamma^C=0=C^0_{IJKL}\\gamma^I\\gamma^K$  \n(so that, using also \\eqref{C_ur_ex}, one gets $C_{abcd}k^bk^d=0$); (ii) $\\bk$ is an mWAND $\\Leftrightarrow C^0_{ABCD}\\gamma^A=0=C^0_{IJKL}\\gamma^I$ (in which case, using also \\eqref{C_ur_ex}, one gets $C_{abcd}k^d=0$). \n\nThese conditions on the Weyl tensors of $\\d\\sigma_\\lambda^2$ and $\\d\\Sigma_0^2$ are generically not satisfied (see an example in section~\\ref{subsubsec_ex2}), so in general spacetimes \\eqref{general_violating} do not admit any other WANDs (not even single) apart from $\\bl$ and $\\bn$. However, it is also clear that in special cases additional mWANDs may exist. For example, when either $\\d\\sigma_\\lambda^2$ or $\\d\\Sigma_0^2$ are conformally flat (which is necessarily the case if $n_\\sigma<4$  or $n_\\Sigma<4$) spacetimes \\eqref{general_violating} admit a continuous infinity of mWANDs (generically non-geodesic but some can be geodesic, see an example in section~\\ref{subsubsec_ex1}). \n\n\n\n\n\\subsection{An explicit example with additional mWANDs}\n\n\n\\label{subsubsec_ex1}\n\nFor the sake of definiteness, one can for instance construct an explicit solution for $n=7$ by taking (\\ref{general_violating}) with \n\\beqn\n & & \\d\\sigma_\\lambda^2=V(\\rho)\\d\\tau^2+V^{-1}(\\rho)\\d\\rho^2+\\rho^2(\\d\\theta^2+\\sin\\theta^2\\d\\phi^2) , \\qquad V(\\rho)=1-\\frac{\\mu}{\\rho}-\\lambda \\rho^2 , \\nonumber \\\\\n & & \\d\\Sigma_0^2=\\d z^2 , \\label{NOCmetric}\n\\eeqn\nand a coordinate range such that $V(\\rho)>0$.\n\nTaking $\\bl=\\pa_r$ and the orthonormal vectors\n\\beqn\n & & \\bbm_{(2)}=A^{-1}V^{-1\/2}(\\rho)\\pa_\\tau , \\qquad \\bbm_{(3)}=A^{-1}V^{1\/2}(\\rho)\\pa_\\rho, \\qquad \\bbm_{(4)}=A^{-1}\\rho^{-1}\\pa_\\theta, \\nonumber \\\\ \n & & \\bbm_{(5)}=A^{-1}\\rho^{-1}\\sin\\theta^{-1}\\pa_\\phi, \\qquad \\bbm_{(6)}=r^{-1}\\pa_z , \\qquad A=\\lambda ur-1 ,\n\\eeqn\none finds\n\\be\n \\rho_{22}=\\rho_{33}=\\rho_{44}=\\rho_{55}=\\frac{\\lambda u}{\\lambda ur-1} , \\qquad \\rho_{66}=\\frac{1}{r} .\n\\ee\n\nThe only non-zero Weyl frame components read (recall definition \\eqref{def_W})\n\\be\n W_{23}=W_{45}=\\frac{1}{(\\lambda ur-1)^2}\\frac{\\mu}{\\rho^3} , \\qquad W_{24}=W_{25}=W_{34}=W_{35}=-\\frac{1}{(\\lambda ur-1)^2}\\frac{\\mu}{2\\rho^3} .\n\\ee\n\nHowever, note  that since $\\d\\Sigma_0^2=\\d z^2$ is (conformally) flat, it follows from section~\\ref{subsubsec_additionalW} that metric \\eqref{general_violating} with \\eqref{NOCmetric} admits also other  mWANDs of the form  \n$\\bk=\\frac{1}{r^2}[\\pa_u-\\frac{1}{2}r^2(\\lambda+\\gamma^2)\\pa_r+\\gamma\\pa_z]$, where $\\gamma$ is an arbitrary function.\n\nIt turns out that if $\\gamma_{,r}=\\gamma_{,u}=\\gamma_{,z}=0$ the mWAND $\\bk$ is geodesic, in which case it becomes twistfree iff $\\gamma$ is a constant and it obeys the optical constraint iff $\\gamma^2=|\\lambda|$ (so that it is also twistfree), i.e.,\n\\be\n\t\\bk=\\frac{1}{r^2}\\left[\\partial_u-\\frac{1}{2}(\\lambda+|\\lambda|)r^2\\pa_r\\pm\\sqrt{|\\lambda|}\\pa_z\\right] .\n\\ee\nIn this case the corresponding optical matrix is of the form $\\{a,a,a,a,a\\}$ for $\\lambda>0$ and $\\{a,a,a,a,0\\}$ for $\\lambda<0$ (in both cases $a\\neq0$).\n\n\n\n\n\\subsection{An explicit example without additional mWANDs}\n\n\n\\label{subsubsec_ex2}\n\n{It follows from section~\\ref{subsubsec_additionalW} that in order for metric \\eqref{general_violating} to admit only two double WANDs we need $n_\\sigma\\ge4$ and $n_\\Sigma\\ge4$, i.e., at least 10 spacetime dimensions. If we now take $\\d\\sigma_\\lambda^2$ as in \\eqref{NOCmetric} but} $\\d\\Sigma_0^2$ corresponding to the Riemannian version of 4d Schwarzschild, {by looking at the conditions of section~\\ref{subsubsec_additionalW} it is easy to see that indeed} the {\\em only} mWANDS are the $\\bl$ and $\\bn$ discussed above (eqs.~\\eqref{l_ex} and \\eqref{n_ex}). Such metric thus constitutes an example of an Einstein spacetime with {\\em all double WANDs violating the optical constraint}. Recall also that such mWANDs \\eqref{l_ex} and \\eqref{n_ex} are geodesic and non-degenerate.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}