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{"text":"\\section{Introduction}\n\\label{intro} In nonequivariant topology, it is a triviality that\nspaces built of only even dimensional cells will have free\ncohomology, regardless of the chosen coefficient ring. It is just\nas easy to see that every space has free cohomology when the\ncoefficient ring is taken to be $\\Z\/2$. Analogous results are not\nso clear in the equivariant setting.\n\nIn \\cite{FL}, it is shown that the $RO(\\Z\/p)$-graded homology of a\n$\\Z\/p$-space built of only even dimensional cells is free as a\nmodule over the homology of a point, regardless of which Mackey\nfunctor is chosen for coefficients. The goal of this paper is to establish a similar result for the cohomology of $G=\\Z\/2$-spaces without the restriction to cells of even degrees, but with the assumption of using\nconstant $\\underline{\\Z\/2}$ Mackey functor coefficients. Here is\nthe main result:\n\n\\begin{thm*}\nIf $X$ is a connected, locally finite, finite\ndimensional $\\text{Rep}(\\Z\/2)$-complex, then\n$H^{*,*}(X;\\underline{\\Z\/2})$ is free as a\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module.\n\\end{thm*}\n\n\\noindent (The bigrading will be explained in Section \\ref{sec:Prelim}.)\n\nThe projective spaces and Grassmann\nmanifolds associated to representations of $\\Z\/2$ are examples of such $\\text{Rep}(\\Z\/2)$-complexes. In these\nparticular cases, the free generators of the cohomology modules are\nin bijective correspondence with the Schubert cells. The precise degrees of the cohomology generators is typically unknown, much\nlike in \\cite{FL}.\n\nSection \\ref{sec:Prelim} provides some of the background and notation required\nfor the rest of the paper. Most of this information can be found in\n\\cite{Alaska} and \\cite{FL} but is reproduced here for convenience.\nSection \\ref{sec:Freeness} holds the main freeness theorem. As\napplications of the freeness theorem, section \\ref{sec:RPs} exhibits\nsome techniques for calculating the cohomology of\n$\\text{Rep}(G)$-complexes. The importance of such calculations lies\nin their potential applications toward understanding $RO(G)$-graded\nequivariant characteristic classes.\n\nThe work in this paper was originally part of the author's\ndissertation while at the University of Oregon.\n\nThe author is indebted to Dan Dugger for his guidance and\ninnumerable helpful conversations.\n\n\n\n\\section{Preliminaries}\n\\label{sec:Prelim} This section contains some of the basic machinery\nand notations that will be used throughout the paper. In this\nsection, $G$ can be any finite group unless otherwise specified.\n\nGiven a $G$-representation $V$, let $D(V)$ and $S(V)$ denote the\nunit disk and unit sphere, respectively, in $V$ with action induced\nby that on $V$. A \\bf$\\text{Rep}(G)$-complex \\rm is a $G$-space $X$ with a\nfiltration $X^{(n)}$ where $X^{(0)}$ is a disjoint union of\n$G$-orbits and $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching\ncells of the form $D(V_\\alpha)$ along maps $f_\\alpha \\colon\nS(V_\\alpha) \\ra X^{(n-1)}$ where $V_\\alpha$ is an $n$-dimensional\nreal representation of $G$. The space $X^{(n)}$ is referred to as\nthe \\bf $n$-skeleton \\rm of $X$, and the filtration is referred to as a \\bf cell\nstructure\\rm .\n\nFor the precise definition of a Mackey functor when $G=\\Z\/2$, the\nreader is referred to \\cite{LMM} or \\cite{DuggerKR}. A summary\nof the important aspects of a Mackey functor is given here. The\ndata of a Mackey functor are encoded in a diagram like the one\nbelow.\n\n\\[\\xymatrix{ M(\\Z\/2) \\ar@(ur,ul)[]^{t^*} \\ar@\/^\/[r]^(0.6){i_*} & M(e) \\ar@\/^\/[l]^(0.4){i^*}} \\]\n\n\nThe maps must satisfy the following four conditions.\n\\begin{enumerate}\n\\item $(t^*)^2 = id$\n\\item $t^*i^*=i^*$\n\\item $i_*(t^*)^{-1}=i_*$\n\\item $i^*i_*=id+t^*$\n\\end{enumerate}\n\nAccording to \\cite{Alaska}, each Mackey functor $M$ uniquely\ndetermines an $RO(G)$-graded cohomology theory characterized by\n\\begin{enumerate}\n\\item $H^n(G\/H;M) =\\begin{cases}\nM(G\/H) & \\text{ if } n=0 \\\\\n0 & \\text{otherwise}\\end{cases}$\n\\item The map $H^0(G\/K;M) \\ra H^0(G\/H;M)$ induced by $i \\colon G\/H \\ra G\/K$ is the transfer map $i^*$ in the Mackey functor.\n\\end{enumerate}\n\nA $p$-dimensional real $\\Z\/2$-representation $V$ decomposes as\n$V=(\\R^{1,0})^{p-q} \\oplus (\\R^{1,1})^q =\\R^{p,q}$ where $\\R^{1,0}$\nis the trivial 1-dimensional real representation of $\\Z\/2$ and $\\R^{1,1}$ is the nontrivial\n1-dimensional real representation of $\\Z\/2$. Thus the $RO(\\Z\/2)$-graded theory is\na bigraded theory, one grading measuring dimension and the other\nmeasuring the number of ``twists''. In this case, we write\n$H^{V}(X;M)=H^{p,q}(X;M)$ for the $V^{\\text{th}}$ graded component\nof the $RO(\\Z\/2)$-graded equivariant cohomology of $X$ with\ncoefficients in a Mackey functor $M$.\n\nIn this paper, $G$ will typically be $\\Z\/2$ and the Mackey functor\nwill almost always be constant $M=\\underline{\\Z\/2}$ which has the\nfollowing diagram.\n\n\\[\\xymatrix{ \\Z\/2 \\ar@(ur,ul)[]^{id} \\ar@\/^\/[r]^{0} & \\Z\/2 \\ar@\/^\/[l]^{id}} \\]\n\n\nWith these constant coefficients, the $RO(\\Z\/2)$-graded cohomology\nof a point is given by the picture in Figure \\ref{fig:pt}.\n\n\\begin{figure}[htpb]\n\\centering\n\\begin{picture}(100,100)(-100,-100)\n\\put(-100,-50){\\vector(1,0){100}}\n\\put(-50,-100){\\vector(0,1){100}}\n\n\n\n\\put(-50, -51){\\line(0,1){40}} \\put(-50, -51){\\line(1,1){40}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\n\\multiput(-90,-51)(20,0){5}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){5}{\\line(1,0){3}}\n\\put(-49,-57){$\\scriptscriptstyle{0}$}\n\\put(-29,-57){$\\scriptscriptstyle{1}$}\n\\put(-9,-57){$\\scriptscriptstyle{2}$}\n\\put(-69,-57){$\\scriptscriptstyle{-1}$}\n\\put(-89,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\n\\put(-48,-35){$\\tau$} \\put(-30,-35){$\\rho$} \\put(-10,-15){$\\rho^2$}\n\\put(-30,-17){$\\tau\\rho$} \\put(-58, -90){$\\theta$} \\put(-48,\n-102){$\\frac{\\theta}{\\tau}$} \\put(-78, -102){$\\frac{\\theta}{\\rho}$}\n\n\\put(-58,0){${q}$} \\put(0,-60){${p}$}\n\n\n\\multiput(-50, -30)(20,20){2}{\\circle*{2}} \\multiput(-50,\n-50)(20,20){3}{\\circle*{2}} \\put(-50, -10){\\circle*{2}} \\put(-50,\n-90){\\circle*{2}} \\put(-50, -110){\\circle*{2}} \\put(-70,\n-110){\\circle*{2}}\n\n\n\\end{picture}\n\n\\caption{$H^{*,*}(pt;\\Z\/2)$} \\label{fig:pt}\n\\end{figure}\n\nEvery lattice point in the picture that is inside the indicated\ncones represents a copy of the group $\\Z\/2$. The \\bf top cone \\rm is a\npolynomial algebra on the nonzero elements $\\rho \\in\nH^{1,1}(pt;\\underline{\\Z\/2})$ and $\\tau \\in\nH^{0,1}(pt;\\underline{\\Z\/2})$. The nonzero element $\\theta \\in H^{0,-2}(pt;\\underline{\\Z\/2})$ in the \\bf bottom\ncone \\rm is infinitely divisible by both $\\rho$ and $\\tau$. The\ncohomology of $\\Z\/2$ is easier to describe:\n$H^{*,*}(\\Z\/2;\\underline{\\Z\/2})=\\Z\/2[t, t^{-1}]$ where $t \\in\nH^{0,1}(\\Z\/2;\\underline{\\Z\/2})$. Details can be found in\n\\cite{DuggerKR} and \\cite{Caruso}.\n\nA useful tool is the following exact sequence of \\cite{AM}.\n\n\\begin{lem}[Forgetful Long Exact Sequence]\n\\label{lemma:forget} Let $X$ be a based $\\Z\/2$-space. Then for\nevery $q$ there is a long exact sequence\n\n$$\\xymatrix{\\cdots \\ar[r] & H^{p,q}(X) \\ar[r]^(0.4){\\cdot \\rho} & H^{p+1,q+1}(X) \\ar[r]^(0.55){\\psi} & H^{p+1}_{sing}(X) \\ar[r]^(0.45)\\delta & H^{p+1,q}(X)} \\ra \\cdots$$\n\n\\end{lem}\n\n\\noindent The map $\\cdot \\rho$ is multiplication by $\\rho \\in\nH^{1,1}(pt;\\underline{\\Z\/2})$ and $\\psi$ is the forgetful map to\nnon-equivariant cohomology with $\\Z\/2$ coefficients.\n\n\n\\section{The Freeness Theorem}\n\\label{sec:Freeness}\n\nComputing the $RO(G)$-graded cohomology of a $G$-space $X$ is\ntypically quite a difficult task. However, if $X$ has a filtration\n$X^{(0)} \\subseteq X^{(1)} \\subseteq \\cdots$, then we can take\nadvantage of the long exact sequences in cohomology arising from the cofiber\nsequences $X^{(n)} \\subseteq X^{(n+1)} \\ra X^{(n+1)}\/X^{(n)}$.\nThese long exact sequences paste together as an exact couple in the usual way,\ngiving rise to a spectral sequence associated to the filtration.\n\nIf $X$ is a $G$-CW complex or a $\\text{Rep}(G)$-complex, then $X$\nhas a natural filtration coming from the cell structure. In either\ncase, if $X$ is connected, the quotient spaces $X^{(n+1)}\/X^{(n)}$\nare wedges of $(n+1)$-spheres with action determined by the type of\ncells that were attached. Examples of this sort appear throughout the paper.\n\nFor the remainder of the paper, we\nwill only be interested in the case $G=\\Z\/2$ and always take coefficients to be $\\underline{\\Z\/2}$. These choices will be implicit in our notation.\n\nGiven a filtered $\\Z\/2$ space $X$, for each fixed $q$ there is a long exact sequence\n\n\\[ \\cdots H^{*,q}(X^{(n+1)}\/X^{(n)})\\ra H^{*,q}(X^{(n+1)}) \\ra H^{*,q}(X^{(n)}) \\ra H^{*+1,q}(X^{(n+1)}\/X^{(n)}) \\cdots \\]\n\n\\noindent and so there is one spectral sequence for each integer $q$. The\nspecifics are given in the following proposition.\n\n\\begin{prop} Let $X$ be a filtered $\\Z\/2$-space. Then for each $q\\in\\Z$ there is a spectral sequence with\n$$E_1^{p,n} = H^{p,q}(X^{(n+1)}, X^{(n)})$$\nconverging to $H^{p,q}(X)$.\n\\end{prop}\n\nThe construction of the spectral sequence is completely standard. See, for example, Proposition 5.3 of \\cite{McC}.\n\nIt is convenient to plot the $RO(\\Z\/2)$-graded cohomology in the\nplane with $p$ along the horizontal axis and $q$ along the vertical\naxis, and this turns out to be a nice way to view the cellular spectral\nsequences as well. When doing so, the\ndifferentials on each page of the spectral sequence have bidegree\n$(1,0)$ in the plane, but reach farther up the filtration on each\npage. It is important to keep track of at\nwhat stage of the filtration each group arises. In practice, this can be done by using different colors for group that arise at different stages of the filtration.\n\nIt is often quite difficult to determine the effect of all of the\nattaching maps in the cell attaching long exact sequences. If $X$\nis locally finite, then the cells can be attached one at a time, in\norder of dimension. This simplicity will make it easier to analyze\nthe differentials in the spectral sequence of the `one at a time'\ncellular filtration, even when the precise impact of the attaching maps are not a priori known.\n\n\\begin{lem}\n Let $B$ be a $\\text{Rep}(\\Z\/2)$-complex with free cohomology that is built only of cells of dimension strictly less than $p$. Suppose $X$ is obtained from $B$ by attaching a single $(p,q)$-cell and let $\\nu$ denote the generator for the cohomology of $X\/B \\cong S^{p,q}$. Then after an appropriate change of basis either\n\\begin{enumerate}\n\\item all attaching maps to the top cone of $\\nu$ are zero (that is, $d(a)=0$ for all $a$ with $a\\in H^{*,q_a}(B)$ with $q_a \\geq q-1$),\n\\item the cell attaching `kills' $\\nu$ and a free generator in dimension $(p-1,q)$, or\n\\item all nonzero differentials hit the bottom cone of $\\nu$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nConsider the cellular spectral sequence associated to attaching a single $(p,q)$ cell to $B$. The effects of attaching such a cell can cause the lower dimensional\ngenerators to hit either the `top cone' or the `bottom cone'\nof the newly attached free generator $\\nu$ of degree $(p,q)$.\n\nSuppose first that all nonzero differentials hit the top\ncone. Then any free generator $\\omega_i$ having a nonzero\ndifferential in the spectral sequence must have degree $(p_i,q_i)$\nwhere $p_i=p-1$ and $q_i \\geq q$. For illustrative purposes, the $E_1$ page of the cellular spectral sequence of an example of this type is pictured in\nFigure \\ref{fig:attach1pqcell}. In this example, there are two generators $\\omega_1$ and $\\omega_2$ with bidegree $(p-1,q_1)$ and one generator $\\omega_3$ with bidegree $(p-1,q_i)$.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-8, 30){\\line(0,1){70}} \\put(-8, 30){\\line(1,1){70}} \\put(-8,\n-10){\\line(0,-1){80}} \\put(-8, -10){\\line(-1,-1){80}} \\put(-8,\n30){\\circle*{2}}\n\\put(-23,25){$\\omega_3$}\n\n\n\\put(-10, 9){\\line(0,1){90}} \\put(-10, 9){\\line(1,1){90}} \\put(-10,\n-31){\\line(0,-1){60}} \\put(-10, -31){\\line(-1,-1){60}} \\put(-10,\n9){\\circle*{2}}\n\n\\put(-40,5){$\\omega_1,\\omega_2$}\n\n\\put(-12, 9){\\line(0,1){90}} \\put(-12, 9){\\line(1,1){90}} \\put(-12,\n-31){\\line(0,-1){60}} \\put(-12, -31){\\line(-1,-1){60}} \\put(-12,\n9){\\circle*{2}}\n\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){130}}\n\\put(10, -71){\\line(0,-1){20}} \\put(10, -71){\\line(-1,-1){20}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(3, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,30){$q_i$} \\put(-60,10){$q_1$}\n\\put(-60,-30){$q$}\n\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence attaching\na single $(p,q)$-cell to $B$.} \\label{fig:attach1pqcell}\n\n\\end{figure}\n\n\nHere, only the generator associated to the $(p,q)$-cell and the\ngenerators with nonzero differentials are shown. Each of the\n$\\omega_i$ satisfies $d(\\omega_i)=\\tau^{n_i} \\nu$ for integers\n$n_i$. Relabeling if necessary, we can arrange so that the\n$\\omega_i$ satisfy $n_1 \\leq n_2 \\leq \\cdots$.\n\nLet $A=\\langle\\omega_i\\rangle$, the $H^{*,*}(pt)$-span of the $\\omega_i$'s. A change of basis can be performed\non $A$, after which we may assume $d(\\omega_1)=\\tau^{n_1}\\nu$ and\n$d(\\omega_i)=0$ for $i>1$. Indeed,\n$\\{\\tau^{n_i-n_1}\\omega_1+\\omega_i\\}$ is a basis for $A$ and\n$d(\\tau^{n_i-n_1}\\omega_1+\\omega_i)=\\tau^{n_1}\\nu$ if $i=1$ and is\nzero otherwise. (In effect, the attaching map can `slide' off of all the $\\omega_i$\nexcept for the one for which $q_i$ is minimal.)\n\nIf $\\omega_1$ happens to be in dimension $(p-1,q)$, then the newly\nattached cell `kills' $\\omega_1$ and $\\nu$.\nOtherwise the nonzero portion of the spectral sequence is illustrated in Figure \\ref{fig:changebasis}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\n\n\\put(-10, 9){\\line(0,1){90}} \\put(-10, 9){\\line(1,1){90}}\n\\put(-10,-31){\\line(0,-1){60}} \\put(-10, -31){\\line(-1,-1){60}}\n\\put(-10, 9){\\circle*{2}} \\put(-17, 5){$\\omega$}\n\n\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){130}}\n\\put(10, -71){\\line(0,-1){20}} \\put(10, -71){\\line(-1,-1){20}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(4, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,10){$q_1$} \\put(-60,-30){$q$}\n\n\\end{picture}\n\\caption{The nonzero portion of the same spectral sequence, after a\nchange of basis.} \\label{fig:changebasis}\n\\end{figure}\n\n\nAfter taking cohomology, the spectral sequence collapses, as in Figure \\ref{fig:E2page}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\n\n\\put(-10, -30){\\line(0,-1){60}} \\put(-10, -30){\\line(-1,-1){60}}\n\\put(-10, -70){\\circle*{2}} \\put(-10, -50){\\circle*{2}}\n\n\\put(-8,-45){$\\omega_1 \\frac{\\theta}{\\tau^m}$}\n\n\\put(10, -30){\\line(0,1){20}} \\put(10,-10){\\line(1,1){110}} \\put(10,\n-30){\\line(1,1){130}} \\put(10, -70){\\line(0,-1){20}} \\put(10,\n-70){\\line(-1,-1){20}} \\put(10, -30){\\circle*{2}}\n\n\\put(14, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,10){$q_1$} \\put(-60,-30){$q$}\n\n\\end{picture}\n\\caption{The $E_2=E_\\infty$ page of the above spectral sequence.}\n\\label{fig:E2page}\n\\end{figure}\n\n\nThere is a class $\\omega_1 \\frac{\\theta}{\\tau^m}$ that, potentially,\ncould satisfy $\\rho \\cdot \\omega_1 \\frac{\\theta}{\\tau^m}=\\nu$.\nHowever, for degree reasons, $\\rho \\cdot \\omega_1\n\\frac{\\theta}{\\tau^{m+1}}=0$ and since $\\rho$ and $\\tau$ commute, $\\rho \\cdot \\omega_1 \\frac{\\theta}{\\tau^m}=0$. This\nmeans $\\nu$ determines a nonzero class in $H^{*,*}(X)$ that is not\nin the image of $\\cdot \\rho$. If $B$ is based, then $X$ is based,\nand so, by the forgetful long exact sequence, $\\nu$ determines a\nnonzero class in non-equivariant cohomology. Then since $\\tau$ maps to $1$\nin non-equivariant cohomology, $\\tau^n \\nu$ is nonzero for all $n$. But,\nas the picture indicates, $\\tau^n \\nu$ is zero for large enough\n$n$. This contradiction implies that there could not have been any\nnonzero differentials hitting the top cone of $\\nu$.\n\nThis argument is independent of whether there are any differentials\nhitting the bottom cone, and so there simply cannot be any nonzero\ndifferentials on the top cone.\\qed\n\\end{proof}\n\n\nDifferentials hitting the bottom cone can cause a shifting in degree of the cohomology generators.\n\n\n\\begin{thm} Suppose $X$ is a $\\text{Rep}(\\Z\/2)$-complex formed by attaching a single $(p,q)$-cell to a space $B$. Suppose also that $\\tilde{H}^{*,*}(B)$ is a free $H^{*,*}(pt)$-module with a single generator $\\omega$ of dimension strictly less than $p$. Then $H^{*,*}(X)$ is a free $H^{*,*}(pt)$-module. In particular, one of the following must hold:\n\\begin{enumerate}\n\\item $H^{*,*}(X) \\cong H^{*,*}(pt)$.\n\n\\item $H^{*,*}(X) \\cong H^{*,*}(B)\\oplus \\Sigma^\\nu H^{*,*}(pt)$, where the degree of $\\nu$ is $(p,q)$.\n\n\\item $H^{*,*}(X)$ is free with two generators $a$ and $b$.\n\\end{enumerate}\n\nIn (3) above, the degrees of the generators $a$ and $b$ are\n$(p-n-1,q-n-1)$ and $(p,q-m-1)$ where\n$d(\\omega)=\\frac{\\theta}{\\rho^n\\tau^m}\\nu$.\n\n\\end{thm}\n\\begin{proof} Under these hypotheses, there is a cofiber sequence of the form $B \\stackrel{i}\\inc X \\stackrel{j}\\fib S^{p,q}$. Denote by $\\nu$ the generator of $H^{*,*}(S^{p,q})$.\n\nIf $d(\\omega)=\\nu$ then $(1)$ holds and $H^{*,*}(X)$ is free. If\n$d(\\omega)=0$, then $(2)$ holds and again $H^{*,*}(X)$ is free. The\nremaining case is $d(\\omega)\\neq0$. By the previous lemma, $d(\\omega)$ is in the bottom cone of $\\nu$. That is $d(\\omega)=\\frac{\\theta}{\\rho^n\\tau^m}\\nu$ for some $n$ and\n$m$. Recall that $\\nu$ has dimension $(p,q)$ and so $\\omega$ has\ndimension $(p-n-1, q-n-m-2)$. The $E_1$ page of the cellular\nspectral sequence is given in Figure \\ref{fig:E1bottom}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\\put(-30, -30){\\line(0,1){130}} \\put(-30, -30){\\line(1,1){130}}\n\\put(-30, -70){\\line(0,-1){20}} \\put(-30, -70){\\line(-1,-1){20}}\n\\put(-30, -30){\\circle*{2}} \\put(-40,-30){$\\omega$}\n\n\\put(10, 71){\\line(0,1){30}} \\put(10, 71){\\line(1,1){30}} \\put(10,\n30){\\line(0,-1){120}} \\put(10, 30){\\line(-1,-1){120}} \\put(10,\n71){\\circle*{2}} \\put(0,70){$\\nu$} \\put(-10, -30){\\circle*{2}}\n\\put(-20,-42){$\\frac{\\theta}{\\rho^n\\tau^m}\\nu$}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\\put(14,-57){$p$} \\put(-60,70){$q$}\n\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence with a\nsingle nonzero differential hitting the bottom cone of an attached\n$(p,q)$-cell.} \\label{fig:E1bottom}\n\\end{figure}\n\n\nAfter taking cohomology, the spectral sequence collapses, and what\nremains is pictured in Figure \\ref{fig:E2bottom}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\\put(-30, 30){\\line(0,1){70}} \\put(-30, 30){\\line(1,1){70}}\n\\put(-30, 30){\\circle*{2}} \\put(-40,30){$a$} \\put(10,\n10){\\line(1,1){90}} \\put(10, 10){\\circle*{2}} \\put(0, 10){$b$}\n\\put(10, 30){\\line(1,1){70}} \\put(10, 30){\\circle*{2}} \\put(0,\n30){$b_1$} \\put(10, 50){\\line(1,1){50}} \\put(10, 50){\\circle*{2}}\n\\put(0, 50){$b_2$} \\put(-30, -70){\\line(0,-1){20}} \\put(-30,\n-70){\\line(-1,-1){20}}\n\n\n\\put(10, 71){\\line(0,1){30}} \\put(10, 71){\\line(1,1){30}} \\put(10,\n71){\\circle*{2}} \\put(-3, 71){$b_m$} \\put(10, -30){\\line(0,-1){60}}\n\\put(10, -30){\\circle*{2}} \\put(-30, -10){\\line(-1,-1){80}}\n\\put(-30, -10){\\circle*{2}} \\put(-30, -30){\\line(-1,-1){60}}\n\\put(-30, -30){\\circle*{2}} \\put(-30, -50){\\line(-1,-1){40}}\n\\put(-30, -50){\\circle*{2}} \\put(-10, -48){\\line(-1,-1){40}}\n\\put(-10, -48){\\line(0,-1){40}} \\put(-10, -48){\\circle*{2}}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\\put(14,-57){$p$} \\put(-60,70){$q$}\n\n\\end{picture}\n\\caption{The $E_2=E_\\infty$ page of the cellular spectral sequence\nwith a single nonzero differential hitting the bottom cone of an\nattached $(p,q)$-cell.} \\label{fig:E2bottom}\n\\end{figure}\n\n\nLet $a$ be the generator in degree $(p-n-1, q-n-1)$ and $b$ the generator in dimension\n$(p,q-m-1)$. For degree reasons, $b$ is not in the image\nof $\\cdot \\rho$ and so determines a nonzero class in non-equivariant\ncohomology. Thus, $\\tau^i b$ is nonzero for all $i$, and so we have\nthat $b_i=\\tau^i b$. In particular, $\\rho^{n+1} a$ and $\\tau^m b$\ngenerate $H^{p,q}(X)$. Consider the portion of the long exact\nsequence associated to the cofiber sequence $B \\stackrel{i}\\inc X\n\\stackrel{j}\\fib S^{p,q}$ given below:\n\n$$\\xymatrix{\\cdots \\ar[r] & H^{p,q}(S^{p,q}) \\ar[r]^{j^*} & H^{p,q}(X) \\ar[r]^{i^*} & H^{p,q}(B) \\ar[r] & 0}$$\n\nSince $i^*(\\rho^{n+1} a)=i^*(\\tau^m b)=\\rho^{n+1}\\tau^m\\omega$,\nexactness implies that $j^*(\\nu)=\\rho^{n+1} a + \\tau^m b$. Also\n$j^*$ is an $H^{*,*}(pt)$-module homomorphism, and so\n$j^*(\\frac{\\theta}{\\rho^{n+1}}\\nu)=\\theta a$ and\n$j^*(\\frac{\\theta}{\\tau^m}\\nu)=\\theta b$. In particular, we can\ncreate a map $f$ from a free module with generators $\\alpha$ and\n$\\beta$ in degrees $(p-n-1,q-n-1)$ and $(p,q-m-1)$ respectively\nto $\\tilde{H}^{p,q}(X)$ with $f(\\alpha)=a$ and $f(\\beta)=b$. This\n$f$ is an isomorphism. \n\n\n\\end{proof}\n\n\\begin{thm}[Freeness Theorem]\n\\label{thm:freeness} If $X$ is a connected, locally finite, finite\ndimensional $\\text{Rep}(\\Z\/2)$-complex, then\n$H^{*,*}(X;\\underline{\\Z\/2})$ is free as a\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module.\n\\end{thm}\n\n\\begin{proof}\n\nSince $X$ is locally finite, the cells can be attached one at a\ntime. Order the cells $\\alpha_1, \\alpha_2, \\dots$ so that their\ndegrees satisfy $p_i \\leq p_j$ if $i\\leq j$ and $q_i \\leq q_j$ if\n$p_i=p_j$ and $i \\leq j$. We can proceed by induction over the\nspaces in the `one-at-a-time' cell filtration $X^{(0)} \\subseteq \\cdots \\subseteq X^{(n)}\n\\subseteq \\cdots \\subseteq X$, with the base case obvious since $X$\nis connected.\n\nFirst, suppose that $H^{*,*}(X^{(n)})$ is a free\n$H^{*,*}(pt)$-module and that $X^{(n+1)}$ is obtained by attaching a\nsingle $(p,q)$-cell and that $X^{(n)}$ has no $p$-cells. Denote by\n$\\nu$ the free generator of $H^{*,*}(X^{(n+1)}\/X^{(n)}) \\cong\nH^{*,*}(S^{p,q})$. Consider the spectral sequence of the filtration\n$X^{(n)} \\subseteq X^{(n+1)}$. An example is pictured below in Figure\n\\ref{fig:free1} to aid in the discussion.\n\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\n\n\\put(-10, -10){\\line(0,1){110}} \\put(-10, -10){\\line(1,1){110}}\n\\put(-10, -50){\\line(0,-1){40}} \\put(-10, -50){\\line(-1,-1){40}}\n\\put(-10, -10){\\circle*{2}}\n\n\\put(-8, 50){\\line(0,1){50}} \\put(-8, 50){\\line(1,1){50}} \\put(-8,\n10){\\line(0,-1){100}} \\put(-8, 10){\\line(-1,-1){100}} \\put(-8,\n50){\\circle*{2}}\n\n\\put(-10, 70){\\line(0,1){30}} \\put(-10, 70){\\line(1,1){30}}\n\\put(-10, 30){\\line(0,-1){120}} \\put(-10, 30){\\line(-1,-1){120}}\n\\put(-10, 70){\\circle*{2}}\n\n\n\\put(-12, -30){\\line(0,1){130}} \\put(-12, -30){\\line(1,1){130}}\n\\put(-12, -70){\\line(0,-1){20}} \\put(-12, -70){\\line(-1,-1){20}}\n\\put(-12, -30){\\circle*{2}}\n\n\\put(-12, -50){\\line(0,1){150}} \\put(-12, -50){\\line(1,1){150}}\n\\put(-12, -50){\\circle*{2}}\n\n\\put(-28, -50){\\line(0,1){150}} \\put(-28, -50){\\line(1,1){150}}\n\\put(-28, -50){\\circle*{2}}\n\n\\put(-48, -70){\\line(0,1){150}} \\put(-48, -70){\\line(1,1){150}}\n\\put(-48, -70){\\circle*{2}}\n\n\\put(-90, -90){\\line(0,1){170}} \\put(-90, -90){\\line(1,1){170}}\n\\put(-90, -90){\\circle*{2}}\n\n\n\\put(10, 31){\\line(0,1){70}} \\put(10, 31){\\line(1,1){70}} \\put(10,\n-10){\\line(0,-1){80}} \\put(10, -10){\\line(-1,-1){80}} \\put(10,\n31){\\circle*{2}} \\put(17,30){$\\nu$} \\put(15,-12){$\\theta\\nu$}\n\n\n\\put(-23,70){$\\alpha'$}\n\\put(-23,50){$\\alpha$}\n\\put(-24,-10){$\\omega'''$}\n\\put(-23,-30){$\\omega''$}\n\\put(-23,-50){$\\omega_n$}\n\\put(-43,-50){$\\omega'$}\n\\put(-63,-70){$\\omega_1$}\n\n\n\n\\end{picture}\n\n\\caption{The spectral sequence of a filtration for attaching a\nsingle $(p,q)$-cell to a space with free cohomology.}\n\\label{fig:free1}\n\\end{figure}\n\nAs before, a change of basis allows us to focus on a subset $\\omega_1, \\dots,\n\\omega_n$ of the free generators of $H^{*,*}(X^{(n)})$ whose\ndifferentials hit the bottom cone of $\\nu$ and that satisfy\n\\begin{enumerate}\n\\item $d(\\omega_i) \\neq 0$ for all $i$,\n\\item $|\\omega_i| > |\\omega_j|$ when $i > j$,\n\\item $|\\omega_i^G| > |\\omega_j^G|$ when $i > j$,\n\\end{enumerate}\n\n\\noindent and all other basis elements have zero differentials to\nthe bottom cone of $\\nu$. This is similar to what is referred to in\n\\cite{FL} as a ramp of length $n$. Also, we can change the basis\nagain so that there is only one free generator, $\\alpha$, of\n$H^{*,*}(X^{(n)})$ with a nonzero differential to the top cone of\n$\\nu$. Then, after this change of basis, the nonzero portion of the\nspectral sequence of the filtration looks like the one in Figure\n\\ref{fig:free2}\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\n\n\\put(-8, 50){\\line(0,1){50}} \\put(-8, 50){\\line(1,1){50}} \\put(-8,\n10){\\line(0,-1){100}} \\put(-8, 10){\\line(-1,-1){100}} \\put(-8,\n50){\\circle*{2}} \\put(-6,45){$\\alpha$}\n\n\\put(-12, -50){\\line(0,1){150}} \\put(-12, -50){\\line(1,1){150}}\n\\put(-12, -50){\\circle*{2}} \\put(-20,-60){$\\omega_{n}$}\n\n\n\n\\put(-48, -70){\\line(0,1){150}} \\put(-48, -70){\\line(1,1){150}}\n\\put(-48, -70){\\circle*{2}} \\put(-55,-80){$\\omega_{n-1}$}\n\n\n\\put(-90, -90){\\line(0,1){170}} \\put(-90, -90){\\line(1,1){170}}\n\\put(-90, -90){\\circle*{2}}\n\n\n\\put(10, 31){\\line(0,1){70}} \\put(10, 31){\\line(1,1){70}} \\put(10,\n-10){\\line(0,-1){80}} \\put(10, -10){\\line(-1,-1){80}} \\put(10,\n31){\\circle*{2}} \\put(17,30){$\\nu$} \\put(15,-12){$\\theta\\nu$}\n\n\n\n\\end{picture}\n\n\\caption{The nonzero portion of the above spectral sequence, after a\nchange of basis.} \\label{fig:free2}\n\\end{figure}\n\nAs above, $\\alpha$ cannot support a nonzero differential, and we can see that each of the\n$\\omega_i$'s will shift up in $q$-degree and $\\nu$ will shift down in $q$-degree. That is, the $\\omega_i$'s and $\\nu$ each give rise to free generators in the cohomology of $H^{*,*}(X^{(n+1)})$, but in different bidegree than their predecessors. Thus, $H^{*,*}(X^{(n+1)})$ is again free.\n\nNow suppose that $X^{(n+1)}$ is obtained by attaching a $(p,q)$-cell\n$\\nu'$ and that $X^{(n)}$ has a single $p$-cell $\\nu$ already. Then\nby the previous case, the generator for $\\nu$ was either shifted\ndown, killed off, or was left alone at the previous stage. In any\ncase, because of our choice of ordering of the cells, the generator\nfor $\\nu$ cannot support a differential to the generator for $\\nu'$.\nThus, the only nonzero differentials to $\\nu'$ are from strictly\nlower dimensional cells. Thus, we are reduced again to the previous\ncase and $H^{*,*}(X^{(n+1)})$ is free. By induction, $H^{*,*}(X)$\nis free. \n\n\\end{proof}\n\n\n\n\\section{Real Projective Spaces and Grassmann Manifolds}\n\\label{sec:RPs}\n\nIn this section, $G=\\Z\/2$ exclusively, and the coefficient Mackey will always be $M = \\underline{\\Z\/2}$ and will be suppressed from the notation.\n\nSince each representation $\\R^{p,q}$ has a linear $\\Z\/2$-action,\nthere is an induced action of $\\Z\/2$ on $G_n(\\R^{p,q})$, the \\bf\nGrassmann manifold \\rm of $n$-dimensional linear subspaces of\n$\\R^{p,q}$. These Grassmann manifolds play a central role in the\nclassification of equivariant vector bundles, and so it is important\nto understand their cohomology. As a special case we have the real\nprojective spaces $\\bP(\\R^{p,q})=G_1(\\R^{p,q})$.\n\nThe usual Schubert cell decomposition endows the Grassmann manifolds\nwith a $\\text{Rep}(\\Z\/2)$-cell structure. However, the number of\ntwists in each cell is dependent upon the flag of subrepresentations\nof $\\R^{p,q}$ that is chosen. A \\bf flag symbol \\rm $\\varphi$ is a sequence\nof integers $\\varphi = (\\varphi_1, \\dots, \\varphi_q)$ satisfying\n$1\\leq \\varphi_1 < \\dots < \\varphi_q \\leq q$. A flag symbol $\\varphi$ determines a flag of subrepresenations $V_0=0 \\subset V_1 \\subset \\cdots\n\\subset V_p=\\R^{p,q}$ satisfying\n$V_{\\varphi_i}\/V_{\\varphi_i-1}=\\R^{1,1}$ for all $i=1, \\dots, q$,\nand all other quotients of consecutive terms are $\\R^{1,0}$. For concreteness, we also require that $V_{i}$ is obtained from $V_{i-1}$ by adjoining a coordinate basis\nvector. For example, there is a\nflag in $\\R^{5,3}$ determined by the flag symbol $\\varphi=(1,3,4)$\nof the form $\\R^{0,0} \\subset \\R^{1,1} \\subset \\R^{2,1} \\subset\n\\R^{3,2} \\subset \\R^{4,3} \\subset \\R^{5,3}$.\n\nA \\bf Schubert symbol \\rm $\\sigma = (\\sigma_1, \\dots , \\sigma_n)$ is a sequence of integers such that $1\\leq\\sigma_1 <\n\\sigma_2 < \\dots < \\sigma_n\\leq p$. Given a Schubert symbol $\\sigma$\nand a flag symbol $\\varphi$, let $e(\\sigma,\\varphi)$ be the set of\nplanes $\\ell \\in G_n(\\R^{p,q})$ for which $\\dim(\\ell \\cap\nV_{\\sigma_i}) = 1+\\dim(\\ell \\cap V_{\\sigma_i-1})$, where $V_0\n\\subset \\cdots \\subset V_n$ is the flag determined by $\\varphi$.\nThen $e(\\sigma,\\varphi)$ is the interior of a cell $D(W)$ for some\nrepresentation $W$. The dimension of the cell is determined by the\nSchubert symbol $\\sigma$ just as in nonequivariant topology, but the\nnumber of twists depends on both $\\sigma$ and the flag symbol\n$\\varphi$.\n\nFor example, consider $G_2(\\R^{5,3})$, $\\sigma = (3,5)$, and\n$\\varphi = (1,3,4)$. Then $e(\\sigma,\\varphi)$ consists of planes\n$\\ell$ which have a basis with echelon form given by the matrix\nbelow.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} \\ast & \\ast & 1 & 0 & 0 \\\\\n\\ast & \\ast & 0 & \\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\nHere, the action of $\\Z\/2$ on the columns, as determined by\n$\\varphi$, has been indicated by inserting the appropriate signs\nabove the matrix. After acting, this becomes the following.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} -\\ast & \\ast & -1 & 0 & 0 \\\\\n-\\ast & \\ast & 0 & -\\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\n\nWe require the last nonzero entry of each row to be 1, and so we\nscale the fisrt row by $-1$.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} \\ast & -\\ast & 1 & 0 & 0 \\\\\n-\\ast & \\ast & 0 & -\\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\nThere are five coordinates which can be any real numbers, three\nof which the $\\Z\/2$ action of multiplication by -1, so this is a\n$(5,3)$-cell. Through a similar process, we can obtain a cell\nstructure for $G_n(\\R^{p,q})$ given any flag $\\varphi$. The type of\ncell determined by the Schubert symbol $\\sigma$ and the flag\n$\\varphi$ is given by the following proposition. Here,\n$\\underline{\\sigma_i} = \\{1, \\dots, \\sigma_i\\}$ and $\\sigma(i) =\n\\{\\sigma_1, \\dots, \\sigma_i\\}$.\n\n\\begin{prop}\n\\label{prop:schubertcells} Let $\\sigma = (\\sigma_1, \\dots,\n\\sigma_n)$ be a Schubert symbol and $\\varphi = (\\varphi_1, \\dots,\n\\varphi_q)$ be a flag symbol for $\\R^{p,q}$. The cell\n$e(\\sigma,\\varphi)$ of $G_n(\\R^{p,q})$ is of dimension $(a,b)$ where\n$a=\\sum_{i=1}^n (\\sigma_i-i)$ and $b=\\sum_{\\sigma_i \\in \\varphi}\n|\\underline{\\sigma_i}\\setminus (\\varphi \\cup\n\\sigma(i))|+\\sum_{\\sigma_i \\notin \\varphi} |(\\underline{\\sigma_i}\n\\cap \\varphi)\\setminus \\sigma(i)|$.\n\n\\end{prop}\n\\begin{proof}\nThe formula for $a$ is exactly the same as in the nonequivariant\ncase. The one for $b$ follows since the number of twisted\ncoordinates in each row is exactly the number of $\\ast$ coordinates\nfor which the action is opposite to that on the coordinate\ncontaining the 1 in that echelon row.\n\\end{proof}\n\n\\begin{cor} Real and complex projective spaces and Grassmann manifolds have free $RO(\\Z\/2)$-graded cohomology with $\\underline{\\Z\/2}$ coefficients.\n\\end{cor}\n\n\\begin{prop} If $V \\subseteq V'$ is an inclusion of representations and $\\varphi \\subseteq \\varphi'$ is an extension of flag symbols for $V$ and $V'$, then there is a cellular inclusion $G_n(V) \\inc G_n(V')$.\n\n\\end{prop}\n\n\nThe following theorem guarantees that the cohomology of Grassmann manifolds have cohomology generators in bijective correspondence with the Schubert cells.\n\n\\begin{thm} $H^{*,*}(G_n(\\R^{u,v}))$ is a free $H^{*,*}(pt)$-module with generators in bijective correspondence with the Schubert cells. \\label{thm:cellbij}\n\\end{thm}\n\\begin{proof}\nSince $G_n(\\R^{u,v})$ has a $\\text{Rep}(\\Z\/2)$-complex structure, we know $H^{*,*}(G_n(\\R^{u,v}))$ is free by the freeness theorem, Theorem \\ref{thm:freeness}. Let $\\{\\omega_1, \\dots,\n\\omega_k\\}$ be a set of free generators. Then $k \\leq m$ where $m$\nis the number of Schubert cells.\n\nThese spaces are based, so we can appeal to the forgetful long exact\nsequence Lemma \\ref{lemma:forget}. By freeness and finite dimensionality,\nthe multiplication by $\\rho$ map\nis an injection for large enough $q$. Thus the forgetful map to\nnon-equivariant cohomology is surjective. Since\n$H^{*}_{sing}(G_n(\\R^{u,v}))$ is free with generators $a_1, \\dots\na_m$ in bijective correspondence with the Schubert cells,\n$H^{*,*}(G_n(\\R^{u,v}))$ has a set of elements, $\\{\\alpha_1, \\dots,\n\\alpha_m\\}$, with $\\psi(\\alpha_i)=a_i$. We can uniquely express\neach $\\alpha_i$ as $\\alpha_i =\\sum_{j=1}^k \\rho^{e_{ij}}\n\\tau^{f_{ij}}\\omega_j$. We can ignore any terms that have $\\rho$ in\nthem since $\\psi(\\rho)=0$. This gives a new set of elements,\n$\\bar{\\alpha}_i = \\sum_{j=1}^k \\epsilon_{ij} \\tau^{f_{ij}}\\omega_j$,\nwhere $\\epsilon_{ij}=0$ or $1$ and $\\psi(\\bar{\\alpha}_i)=a_i$. Since\n$\\psi(\\tau)=1$, we have that $\\sum_{j=1}^k\n\\epsilon_{ij}\\psi(\\omega_j)=a_i$. Since linear combinations of the\nlinearly independent $\\omega_j$'s map to the linearly independent\n$a_i$'s, there are at least as many $\\omega_j$'s as there are\n$a_i$'s. That is, $k \\geq m$. \n\\end{proof}\n\nThe above theorem is enough to determine the additive structure of the $RO(\\Z\/2)$-graded cohomology of the real projective spaces.\n\nRecall that ${\\cat U}=(\\R^{2,1})^\\infty$ is a complete universe in the sense of \\cite{Alaska}. Denote by\n$\\R\\bP^\\infty_{tw}=\\bP({\\cat U})$, the space of lines in the\ncomplete universe ${\\cat U}$.\n\nDenote by $\\R\\bP^{n}_{tw}=\\bP(\\R^{n+1,\\left\\lfloor \\frac{n+1}{2}\n\\right\\rfloor})$, the equivariant space of lines in\n$\\R^{n+1,\\left\\lfloor \\frac{n+1}{2} \\right\\rfloor}$. For example,\n$\\R\\bP^3_{tw} = \\bP(\\R^{4,2})$, $\\R\\bP^4_{tw} = \\bP(\\R^{5,2})$, and\n$\\R\\bP^1_{tw} = S^{1,1}$. There are natural cellular inclusions\n$\\R\\bP^n_{tw} \\inc \\R\\bP^{n+1}_{tw}$, the colimit of which is $\\R\\bP^\\infty_{tw}$.\n\n\n\n\n\\begin{lem} $\\R \\bP^{n}_{tw}$ has a $\\text{Rep}(\\Z\/2)$-structure with cells in dimension $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2)$, $\\dots, (n,\\left\\lceil \\frac{n}{2} \\right\\rceil)$.\n\\end{lem}\n\\begin{proof} This follows from Proposition \\ref{prop:schubertcells} using the flag symbol $\\varphi=(2, 4, 6, \\dots)$. \n\\end{proof}\n\n\\begin{lem}\n$\\R \\bP^\\infty_{tw}$ has a cell\nstructure with a single cell in dimension $(n, \\left\\lceil\n\\frac{n}{2} \\right\\rceil)$, for all $n \\in \\N$.\n\\end{lem}\n\\begin{proof}\nThe inclusions $\\R\\bP^{1}_{tw} \\inc \\R\\bP^{2}_{tw} \\inc\n\\cdots$ are cellular and their colimit is $\\R\\bP^\\infty_{tw}$. \n\\end{proof}\n\n\n\\begin{prop}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(\\R \\bP^{n}_{tw})$ is free with a\nsingle generator in each degree $(k, \\left\\lceil \\frac{k}{2}\n\\right\\rceil)$ for $k= 0, 1, \\dots, n$.\n\n\\end{prop}\n\\begin{proof}\nAny nonzero differentials in the cellular spectral sequence associated to the cell structure using the flag symbol $\\varphi = (2,4,6,\\dots)$ would decrease the number of cohomology generators below the number of cells. (See Figures \\ref{fig:nodd} and \\ref{fig:neven}.) By Theorem \\ref{thm:cellbij} this is not the case, and so the cohomology generators have degrees matching the dimensions of the cells. \n\\end{proof}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{picture}(330,230)(-150,-110)\n\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}} \\put(-26,-35){$a_{1,1}$}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}} \\put(-6,-35){$a_{2,1}$}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\\put(52, 10){\\line(0,1){70}} \\put(52, 10){\\line(1,1){50}} \\put(52,\n-30){\\line(0,-1){80}} \\put(52, -30){\\line(-1,-1){80}} \\put(52,\n10){\\circle*{2}} \\put(55,7){$a_{n,(n-1)\/2}$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(115,-60){${p}$}\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$\\R\\bP^{n}_{tw}$ for $n$ odd.} \\label{fig:nodd}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}} \\put(-26,-35){$a_{1,1}$}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}} \\put(-6,-35){$a_{2,1}$}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\\put(52, 10){\\line(0,1){70}} \\put(52, 10){\\line(1,1){50}} \\put(52,\n-30){\\line(0,-1){80}} \\put(52, -30){\\line(-1,-1){80}} \\put(52,\n10){\\circle*{2}}\n\n\\put(70, 10){\\line(0,1){70}} \\put(70, 10){\\line(1,1){30}} \\put(70,\n-30){\\line(0,-1){80}} \\put(70, -30){\\line(-1,-1){80}} \\put(70,\n10){\\circle*{2}} \\put(73,7){$a_{n,n\/2}$}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(115,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$\\R\\bP^{n}_{tw}$ for $n$ even.} \\label{fig:neven}\n\\end{figure}\n\n\n\n\n\\begin{prop}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(\\R \\bP^\\infty_{tw})$ is free\nwith a single generator in each degree $(n, \\left\\lceil \\frac{n}{2}\n\\right\\rceil)$, for all $n \\in \\N$.\n\n\\end{prop}\n\n\\begin{proof}\n\n$\\R\\bP^\\infty_{tw}$ is the colimit of the above projective spaces.\nThus, any non-zero differential for $\\R\\bP^\\infty_{tw}$ would induce\na non-zero differential at some finite stage. By the above proposition, this is not the case. \n\n\\end{proof}\n\n\\begin{lem}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(S^{1,1})$ is free with a single\ngenerator $a$ in degree $(1,1)$. As a ring, $H^{*,*}(S^{1,1}) \\cong\nH^{*,*}(pt)[a]\/(a^2 = \\rho a)$.\n\n\\end{lem}\n\\begin{proof}\n\nThe statement about the module structure is immediate since $S^{1,1}\n\\cong \\R\\bP^1_{tw}$.\n\nSince $S^{1,1}$ is a $K(\\Z(1),1)$, we can consider $a \\in\n[S^{1,1},S^{1,1}]$ as the class of the identity and $\\rho \\in [pt,\nS^{1,1}]$ as the inclusion. Then $a^2$ is the composite\n\n$\\xymatrix{a^2 \\colon S^{1,1} \\ar[r]^(0.45)\\Delta & S^{1,1} \\Smash\nS^{1,1} \\ar[r]^(0.6){a \\Smash a} & S^{2,2} \\ar[r] & K(\\Z\/2(2),2)}$.\n\n\\noindent Similarly, $\\rho a$ is the composite\n\n$\\xymatrix{\\rho a \\colon S^{1,1} \\ar[r] & S^{0,0} \\Smash S^{1,1}\n\\ar[r]^(0.6){\\rho \\Smash a} & S^{2,2} \\ar[r] & K(\\Z\/2(2),2)}$.\n\n\\noindent The claim is that these two maps are homotopic.\nConsidering the spheres involved as one point compactifications of\nthe corresponding representations, the map $a^2$ is inclusion of\n$(\\R^{1,1})^+$ as the diagonal in $(\\R^{2,2})^+$ and $\\rho a$ is\ninclusion of $(\\R^{1,1})^+$ as the vertical axis. There is then an\nequivariant homotopy $H\\colon (\\R^{1,1})^+ \\times I \\ra\n(\\R^{2,2})^+$ between these two maps given by $H(x,t) = (t x, x)$.\n\n\n\\end{proof}\n\n\nFrom here, we are poised to compute the ring structure of the $RO(\\Z\/2)$-graded cohomology of each real projective space.\n\n\n\\begin{thm}\n\\label{thm:rpinfty} $H^{*,*}(\\R\\bP^\\infty_{tw})=H^{*,*}(pt)[a,\nb]\/(a^2=\\rho a +\\tau b)$, where $\\deg(a)=(1,1)$ and $\\deg(b)=(2,1)$.\n\\end{thm}\n\\begin{proof}\nIt remains to compute the multiplicative\nstructure of the cohomology ring. Denote by $a = a_{(1, 1)}$, and $b=a_{(2,1)}$.\nBy Lemma \\ref{lemma:forget}, the forgetful map $\\psi\\colon H^{*,*}(\\R\n\\bP^\\infty_{tw}) \\ra\nH_{sing}^*(\\R\\bP^\\infty)$ maps $\\psi(a) = z$ and $\\psi(b) = z^2$\nwhere $z \\in H^1_{sing}(\\R\\bP^\\infty)$ is the ring generator for\nnon-equivariant cohomology. Since $\\psi$ is a homomorphism of rings,\n$\\psi(ab)=z^3 \\neq 0$, and so the product $ab$ is nonzero in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$.\nObserve that $\\rho b$ is also in degree $(3,2)$ in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$, but\n$\\psi(\\rho b) = 0$ since $\\psi(\\rho)=0$. Thus $ab$ and $\\rho b$\ngenerate $H^{*,*}(\\R\n\\bP^\\infty_{tw})$ in degree $(3, 2)$. Also, $\\psi(b^2)=z^4$, and so\n$b^2$ in nonzero in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$. This means that $b^2$ is the unique\nnonzero element of $H^{*,*}(\\R\n\\bP^\\infty_{tw})$ in degree $(4,2)$. Inductively, it can be\nshown that if $n$ is even the unique nonzero element of $R$ in\ndegree $(n,\\frac{n}{2})$ is $b^{n\/2}$ and that if $n$ is odd, then\n$ab^{(n-1)\/2}$ is linearly independent from $\\rho b^{(n-1)\/2}$.\n\nNow, $a^2 \\in H^{2,2}(\\R \\bP^\\infty_{tw})$ and so is a linear\ncombination of $\\rho a$ and $\\tau b$. Since $\\psi(a^2)=z^2$, there\nmust be a $\\tau b$ term in the expression for $a^2$. Also, upon\nrestriction to $\\R\\bP^1_{tw}=S^{1,1}$, $a^2$ restricts to $a^2=\\rho\na$. Thus, $a^2=\\rho a + \\tau b \\in H^{*,*}(\\R\n\\bP^\\infty_{tw})$.\n\n\n\n\\end{proof}\n\n\\begin{thm} Let $n > 2$. If $n$ is even, then $H^{*,*}(\\bP(\\R^{n,\\frac{n}{2}})) = H^{*,*}(pt)[a_{1,1},b_{2,1}]\/ \\sim$ where the generating relations are $a^2 = \\rho a + \\tau b$ and $b^k =0$ for $k \\geq \\frac{n}{2}$. If $n$ is odd, then $H^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}})) = H^{*,*}(pt)[a_{1,1},b_{2,1}]\/ \\sim$ where the generating relations are $a^2 = \\rho a + \\tau b$, $b^k =0$ for $k \\geq \\frac{n+1}{2}$, and $a\\cdot b^{(n-1)\/2}=0$.\n\n\\end{thm}\n\n\\begin{proof}\nOnly the multiplicative structure needs to be checked since the\ncohomology is free and the generators given above are in the correct\ndegrees. Considering the restriction of the corresponding\nclasses $a$ and $b$ in $H^{*,*}(\\R\\bP^\\infty_{tw})$, the relation\n$a^2 = \\rho a + \\tau b$ is immediate. The relations $b^k =0$ for $k\n> \\frac{n}{2}$ when $n$ is even and $b^k =0$ for $k \\geq\n\\frac{n+1}{2}$ when $n$ is odd follow for degree reasons. Also,\nsince the class $ab^{(n-1)\/2} \\in H^{*,*}(\\R\\bP^\\infty_{tw})$ is a free\ngenerator, it restricts to zero in\n$H^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}}))$. Thus $ab^{(n-1)\/2} = 0 \\in\nH^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}}))$.\n\n\\end{proof}\n\n\nWe can also compute the cohomology of projective spaces associated\nto arbitrary representations. The following easy lemma will be\nuseful. In particular, it allows us to only consider the projective\nspaces associated to representations $V \\cong \\R^{p,q}$ where $q\\leq\np\/2$.\n\n\\begin{lem}\n\\label{lemma:easyRP} $\\bP(\\R^{p,q}) \\cong \\bP(\\R^{p,p-q})$.\n\n\n\\end{lem}\n\\begin{proof}\nConsider a basis of $\\R^{p,q}$ in which the first $q$ coordinates\nhave the nontrivial action, and a basis of $\\R^{p,p-q}$ in which the\nfirst $q$ coordinates are fixed by the action. Then the map $f\n\\colon \\bP(\\R^{p,q}) \\ra \\bP(\\R^{p,p-q})$ that sends the span of\n$(x_1, \\dots , x_p)$ to the span of $(x_1, \\dots , x_p)$ is\nequivariant. It is clearly a homeomorphism. \n\\end{proof}\n\n\\begin{lem}If $q \\leq p\/2$, then $\\bP(\\R^{p,q})$ has a cell structure with a single cell in each dimension $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2), \\dots,$ $(2q-1,q)$, $(2q,q)$, $\\dots,$ $(p-1,q)$.\n\\end{lem}\n\nFor example, $\\bP(\\R^{4,1})$ has a single cell in each dimension\n$(0,0)$, $(1,1)$, $(2,1)$, and $(3,1)$.\n\n\\begin{proof}\nThe result follows by Proposition \\ref{prop:schubertcells} using the flag symbol $\\varphi=(2,4,\\dots,2q)$. \n\n\\end{proof}\n\n\\begin{lem}\n\\label{lemma:RPmodule} As a $H^{*,*}(pt)$-module,\n$H^{*,*}(\\bP(\\R^{p,q}))$ is free with a single generator in\ndegrees $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2), \\dots,$\n$(2q,q)$, $(2q+1,q), \\dots,$ $(p-1,q)$.\n\n\\end{lem}\n\n\\begin{proof}\n\nUsing the cell structure in the previous lemma, Theorem \\ref{thm:cellbij} implies there can be no nonzero differentials in the cellular spectral sequence.\n\n\\end{proof}\n\n\nThe ring structure of the other projective spaces can be computed by considering the restriction of\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ to $H^{*,*}(\\bP(\\R^{p,q}))$.\n\\begin{prop}\n\\label{prop:ringRP} $H^{*,*}(\\bP(\\R^{p,q}))$ is a truncated\npolynomial algebra over $H^{*,*}(pt)$ on generators in degrees\n$(1,1)$, $(2,1)$, $(2q+1,q)$, $(2q+2,q), \\dots,$ $(p-1,q)$, subject\nto the relations determined by the restriction of\n\n\n\n\n\n\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ to $H^{*,*}(\\bP(\\R^{p,q}))$.\n\n\\end{prop}\n\nFor example, consider $\\bP(\\R^{4,1})$. By the above proposition, $H^{*,*}(\\bP(\\R^{4,1}))$ is generated by classes\n$a_{1,1}$, $b_{2,1}$, and $c_{3,1}$. The classes $a$\nand $b$ in $H^{*,*}(\\R\\bP^\\infty_{tw})$ restrict to $a$ and $b$\nrespectively, so $a^2 = \\rho a + \\tau b$ in\n$H^{*,*}(\\bP(\\R^{4,1}))$. Now, $ab$ has degree $(3,2)$ and so $ab =\n? \\rho b + ? \\tau c$. However, the product $ab$ in\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ restricts to the class $\\tau c$. Since\nrestriction is a map of rings, $ab = \\tau c$ in\n$H^{*,*}(\\bP(\\R^{4,1}))$. Similar considerations show that $bc =0$\nand $c^2=0$. Thus $H^{*,*}(\\bP(\\R^{4,1}))=H^{*,*}(pt)[a_{1,1},\nb_{2, 1}, c_{3,1}]\/\\sim$, where the generating relations are $a^2 =\n\\rho a + \\tau b$, $ab = \\tau c$, $bc=0$, and $c^2=0$.\n\nIn some cases, the Freeness Theorem is enough to determine the\nadditive structure of the $RO(\\Z\/2)$-graded cohomology of Grassmann\nmanifolds.\n\n\\begin{prop} $G_2(\\R^{p,1})$ has a $\\text{Rep}(\\Z\/2)$-complex structure so that\n$H^{*,*}(G_n(\\R^{p,1}))$ is a free $H^{*,*}(pt)$-module on\ngenerators whose degree match the dimensions of the cells.\n\\end{prop}\n\n\\begin{proof}\nUsing the flag symbol $\\varphi=(2)$, every cell, except the\n$(0,0)$-cell, has either one or two twists. The cells are in\nbidegrees so that there can be no dimension shifting in the cellular\nspectral sequence. The result now follows by Theorem\n\\ref{thm:cellbij}. \n\\end{proof}\n\nFor example, $H^{*,*}(G_2(\\R^{4,1});\\underline{\\Z\/2})$ is a free\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module with generators in degrees\n$(0,0)$, $(1,1)$, $(2,1)$, $(2,1)$, $(3,1)$, and $(4,2)$ (see Figure\n\\ref{fig:g2r41}).\n\n\n\n\\begin{figure}[htbp]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){70}} \\put(-50, -51){\\line(1,1){150}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){130}} \\put(-28, -31){\\line(1,1){130}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){130}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){130}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){90}}\n\\put(10, -71){\\line(0,-1){40}} \\put(10, -71){\\line(-1,-1){40}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){110}} \\put(32, -11){\\line(1,1){70}}\n\\put(32, -51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}}\n\\put(32, -11){\\circle*{2}}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{$H^{*,*}(G_2(\\R^{4,1}))$} \\label{fig:g2r41}\n\\end{figure}\n\n\nInterestingly, there are situations where there must be nonzero\ndifferentials in the cellular spectral sequences.\n\n\nAs another example, consider now $X=G_2(\\R^{4,2})$. Consider the\nthree flag symbols $\\varphi_1=(2,3)$, $\\varphi_2=(2,4)$, and\n$\\varphi_3=(3,4)$. The spectral sequences associated to the cell\nstructures with these flag symbols have $E_1$ term given in Figures\n\\ref{fig:g2r42phi23}, \\ref{fig:g2r42phi24}, and \\ref{fig:g2r42phi34}\nrepsectively.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -51){\\line(0,1){130}} \\put(-28, -51){\\line(1,1){130}}\n\\put(-28, -91){\\line(0,-1){20}} \\put(-28, -91){\\line(-1,-1){20}}\n\\put(-28, -51){\\circle*{2}}\n\n\\put(-10, -11){\\line(0,1){90}} \\put(-10, -11){\\line(1,1){90}}\n\\put(-10, -51){\\line(0,-1){60}} \\put(-10, -51){\\line(-1,-1){60}}\n\\put(-10, -11){\\circle*{2}}\n\n\\put(-8, -11){\\line(0,1){90}} \\put(-8, -11){\\line(1,1){90}} \\put(-8,\n-51){\\line(0,-1){60}} \\put(-8, -51){\\line(-1,-1){60}} \\put(-8,\n-11){\\circle*{2}}\n\n\\put(10, -11){\\line(0,1){90}} \\put(10, -11){\\line(1,1){90}} \\put(10,\n-51){\\line(0,-1){60}} \\put(10, -51){\\line(-1,-1){60}} \\put(10,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){90}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_1=(2,3)$.} \\label{fig:g2r42phi23}\n\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){110}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, 10){\\line(0,1){70}} \\put(10, 10){\\line(1,1){70}} \\put(10,\n-31){\\line(0,-1){80}} \\put(10, -31){\\line(-1,-1){80}} \\put(10,\n10){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){90}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_2=(2,4)$.} \\label{fig:g2r42phi24}\n\n\\end{figure}\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){110}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, -31){\\line(0,1){110}} \\put(10, -31){\\line(1,1){110}}\n\\put(10, -71){\\line(0,-1){40}} \\put(10, -71){\\line(-1,-1){40}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(32, 29){\\line(0,1){50}} \\put(32, 29){\\line(1,1){50}} \\put(32,\n-11){\\line(0,-1){100}} \\put(32, -11){\\line(-1,-1){100}} \\put(32,\n29){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_3=(3,4)$.} \\label{fig:g2r42phi34}\n\n\\end{figure}\n\nThe cohomology of $X$ can be deduced by comparing these three\ncellular spectral sequences. We can see from the picture for\n$\\varphi_2$ that $H^{1,0}(X)=0$, and so the differential leaving the\n$(1,0)$ generator in the $\\varphi_1$ spectral sequence is non-zero.\nThus, $H^{1,1}(X)=\\Z\/2$, $H^{2,1}(X)=\\Z\/2$ and $H^{2,0}(X)=\\Z\/2$.\nIn particular, there is a free generator in degree $(1,1)$ and there\nis a nontrivial differential leaving the $(2,1)$ generators of the\nspectral sequence for $\\varphi_2$. After a change of basis, if\nnecessary, the differential can be adjusted so that it is zero on\none of the $(2,1)$ generators and the other generator maps\nnontrivially. Now from $\\varphi_1$ we see that $H^{4,1}(X)=0$, and\nso there is a nontrivial differential leaving the $(3,1)$ generator\nin the $\\varphi_3$ spectral sequence. This means that the $(4,2)$\ngenerator in the $\\varphi_1$ and $\\varphi_2$ spectral sequences must\nsurvive. Thus, all differentials in the $\\varphi_2$ spectral\nsequence are known. They are all zero, except for the one leaving\nthe two $(2,1)$ generators, which behaves as described above. That\nspectral sequence collapses almost immediately to give the\ncohomology of $G_2(\\R^{4,2})$ pictured in Figure \\ref{fig:g2r42}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -13){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(-8,\n-71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}} \\put(-8,\n-13){\\circle*{2}} \\put(12, -11){\\circle*{2}}\n\n\\put(10, 10){\\line(0,1){70}} \\put(10, 10){\\line(1,1){70}} \\put(10,\n-51){\\line(0,-1){60}} \\put(-10, -51){\\line(-1,-1){40}} \\put(10,\n10){\\circle*{2}} \\put(10, -51){\\circle*{2}} \\put(-10,\n-51){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{$H^{*,*}(G_2(\\R^{4,2}))$} \\label{fig:g2r42}\n\\end{figure}\n\nBy the Freeness Theorem \\ref{thm:freeness}, we know that\n$H^{*,*}(G_2(\\R^{4,2}))$ is free. Counting the $\\Z\/2$ vector space\ndimensions in each bidegree reveals that the degrees are the same as\nthose of a free $H^{*,*}(pt)$-module with generators in degrees\n$(1,1)$, $(2,1)$, $(2,2)$, ($3,2)$, and $(4,2)$. This is the only\nfree $H^{*,*}(pt)$-module with these $\\Z\/2$ dimensions, and so we\nhave the following computation.\n\n\\begin{prop}$H^{*,*}(G_2(\\R^{4,2}))$ is a free $H^{*,*}(pt)$-module with generators in degrees $(1,1)$, $(2,1)$, $(2,2)$, $(3,2)$, and $(4,2)$.\n\\end{prop}\n\nThat is, $H^{*,*}(G_2(\\R^{4,2}))$ has free generators as displayed\nin Figure \\ref{fig:cohomg2r42}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -11){\\line(0,1){90}} \\put(-8, -11){\\line(1,1){90}} \\put(-8,\n-51){\\line(0,-1){60}} \\put(-8, -51){\\line(-1,-1){60}} \\put(-8,\n-11){\\circle*{2}}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(30, -11){\\line(0,1){90}} \\put(30, -11){\\line(1,1){70}} \\put(30,\n-51){\\line(0,-1){60}} \\put(30, -51){\\line(-1,-1){60}} \\put(30,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\n\\caption{$H^{*,*}(G_2(\\R^{4,2}))$ with free generators shown.}\n\\label{fig:cohomg2r42}\n\\end{figure}\n\nIt should be noted that in the case of $G_2(\\R^{4,1})$, with the\nproper choice of flag symbols, the cell structure is such that the\ndifferentials are all zero, and so the cohomology is free with\ngenerators in the same degrees as the dimensions of the cells. This\nis \\bf{not} \\rm the case with $G_2(\\R^{4,2})$. Regardless of the\nchoice of flag symbol, there are some nonzero differentials which\ncause some degree shifting of the cohomology generators.\n\nUnfortunately, we cannot play this game indefinitely. For the\nGrassmann manifolds $G_n (\\R^{p,q})$ with $n$ and $q$ small enough,\nsay $n\\leq 2$ and $q\\leq 2$, the above techniques can be used to\nobtain the additive structure of $H^{*,*}(G_n (\\R^{p,q}))$. However,\nthere are examples where the precise degrees of the cohomology\ngenerators cannot be determined by comparing the cellular spectral\nsequences for various flag symbols. A serious inquiry into the\ngeometry of the attaching maps in these cell structures may reveal\nmore information.\n\\newpage\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} |
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{"text":"\\section{Introduction} \n\\setcounter{equation}{0}\nOn a compact $\\Spin$ surface, Th. Friedrich and E.C. Kim proved that any eigenvalue $\\lambda$ of the Dirac operator satisfies the equality \\cite[Thm. 4.5]{fk}:\n\\begin{equation}\\label{eq:Fk}\n\\lambda^2=\\frac{\\pi\\chi(M)}{Area(M)}+\\frac{1}{Area(M)}\\int_M|T^\\psi|^2v_g, \n\\end{equation}\nwhere $\\chi(M)$ is the Euler-Poincar\\'e characteristic of $M$ and $T^\\psi$ is the field of quadratic forms called the Energy-Momentum tensor. It is given on the complement set of zeroes of the eigenspinor $\\psi$ by \n$$ T^\\psi (X, Y) = g(\\ell^\\psi (X), Y) = \\frac 12 \\Re(X\\cdot\\nabla_Y\\psi + Y\\cdot\\nabla_X\\psi,\\frac{\\psi}{\\vert\\psi\\vert^2}),$$\nfor every $X, Y \\in \\Gamma(TM)$. Here $\\ell^\\psi$ is the field of symmetric endomorphisms associated with the field of quadratic forms $T^\\psi$. We should point out that since $\\psi$ is an eigenspinor, the zero set is discret \\cite{barrr}.\nThe proof of Equality (\\ref{eq:Fk}) relies mainly on a local expression of the covariant derivative of $\\psi$ and the use of the Schr\\\"odinger-Lichnerowicz formula. This equality has many direct consequences.\nFirst, since the trace of $\\ell^\\psi$ is equal to $\\lambda$, we have by the Cauchy-Schwarz inequality that $\\vert \\ell^\\psi\\vert^2 \\geqslant \\frac{(tr(\\ell^\\psi))^2}{n} = \\frac{\\lambda^2}{2},$ where $tr$ denotes the trace of $\\ell^\\psi$. Hence, Equality (\\ref{eq:Fk}) implies the B\\\"{a}r inequality \\cite{bar2} given by\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\lambda_1^2 := \\frac{2\\pi\\chi(M)}{Area(M)}.\n \\label{barspin}\n\\end{eqnarray}\nMoreover, from Equality (\\ref{eq:Fk}), Th. Friedrich and E.C. Kim \\cite{fk} deduced\n that $\\int_M {\\rm det}(T^\\psi)v_g=\\pi\\chi(M)$, which gives an information on the Energy-Momentum tensor without knowing the eigenspinor nor the eigenvalue. Finally, for any closed surface $M$ in $\\mathbb{R}^3$ of constant mean curvature $H$, the restriction to $M$ of a parallel spinor on $\\mathbb{R}^3$ is a generalized Killing spinor of eigenvalue $-H$ with Energy-Momentum tensor equal to the Weingarten tensor $II$ (up to the factor $-\\frac{1}{2})$ \\cite{m1} and we have by \\eqref{eq:Fk}\n$$H^2=\\frac{\\pi\\chi(M)}{Area(M)}+\\frac{1}{4Area(M)}\\int_M|II|^2v_g.$$\nIndeed, given any surface $M$ carrying such a spinor field, Th. Friedrich \\cite{fr3} showed \nthat the Energy-Momentum tensor associated with this spinor satisfies the Gauss-Codazzi equations and hence $M$ is locally immersed into $\\mathbb{R}^3$.\\\\\\\\\nHaving a $\\Spinc$ structure on manifolds is a weaker condition than having a $\\Spin$ structure because every $\\Spin$ manifold has a trivial $\\Spinc$ structure. Additionally, any compact surface or any product of a compact surface with $\\mathbb{R}$ has a $\\Spinc$ structure carrying particular spinors. In the same spirit as in \\cite{13}, when using a suitable conformal change, the second author \\cite{r1} established a B\\\"{a}r-type inequality for the eigenvalues of the Dirac operator on a compact surface endowed with any $\\Spinc$ structure. In fact, any eigenvalue $\\lambda$ of the Dirac operator satisfies\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\lambda_1^2 := \\frac{2\\pi \\chi(M)}{Area(M)} - \\frac{1}{Area(M)} \\int_M \\vert\\Omega\\vert v_g,\n\\label{barspinc}\n\\end{eqnarray}\nwhere $i\\Omega$ is the curvature form of the connection on the line bundle given by the $\\Spinc$ structure.\nEquality is achieved if and only if the eigenspinor $\\psi$ associated with the first eigenvalue $\\lambda_1$ is a Killing $\\Spinc$ spinor, i.e., for every $X\\in \\Gamma(TM)$ the eigenspinor $\\psi$ satisfies\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\nabla_X\\psi = -\\frac{\\lambda_1}{2}X\\cdot\\psi,\\\\\n\\Omega\\cdot\\psi =i \\vert\\Omega\\vert\\psi.\n\\end{array}\n\\label{lamba}\n\\right.\n\\end{eqnarray}\nHere $X\\cdot\\psi$ denotes the Clifford multiplication and $\\nabla$ the spinorial Levi-Civita connection \\cite{fr1}.\\\\\\\\\nStudying the Energy-Momentum tensor on a compact Riemannian $\\Spin$ or $\\Spinc$\nmanifolds has been done by many authors, since it is related to several geometric situations.\nIndeed, on compact $\\Spin$ manifolds, J.P. Bourguignon and P. Gauduchon \\cite{BG92}\nproved that the Energy-Momentum tensor appears naturally in the study of the variations\nof the spectrum of the Dirac operator. Th. Friedrich and E.C. Kim \\cite{fk2} obtained\nthe Einstein-Dirac equation as the Euler-Lagrange equation of a certain functional. The second author extented \nthese last two results to $\\Spinc$ manifolds \\cite{r2}. Even if it is not a\ncomputable geometric invariant, the Energy-Momentum tensor is, up to a constant, the\nsecond fundamental form of an isometric immersion into a $\\Spin$ or $\\Spinc$ manifold carrying a\nparallel spinor \\cite{m1, r2}. For a better understanding of the tensor $q^\\phi$\n associated with a spinor field $\\phi$, the first author \\cite{habib1}\n studied Riemannian flows and proved\nthat, if the normal bundle carries a parallel spinor $\\psi$, the tensor $q^\\phi$ associated with $\\phi$ (the restriction of $\\psi$ to the flow) \n is the O'Neill tensor of the flow.\\\\\\\\\nIn this paper, we give a formula corresponding to \\eqref{eq:Fk} for any eigenspinor $\\psi$ of the square of the Dirac operator on compact surfaces endowed with any $\\Spinc$ structure (see Theorem \\ref{procarrdir}). It is motivated by the following two facts: First, when we consider eigenvalues of the square of the Dirac operator, another tensor field is of interest. It is the skew-symmetric tensor field $Q^\\psi$ given by \n$$Q^\\psi(X,Y)= g(q^\\psi (X), Y) = \\frac{1}{2}\\Re(X\\cdot\\nabla_Y\\psi-Y\\cdot\\nabla_X\\psi,\\frac{\\psi}{|\\psi|^2}),$$ \nfor all vector fields $X,Y\\in \\Gamma(TM).$ This tensor was studied by the first author in the context of Riemannian flows \\cite{habib1}. Second, we consider any compact surface $M$ immersed in $\\cercle^2\\times \\mathbb{R}$ where $\\cercle^2$ is the round sphere equipped with a metric of curvature one. The $\\Spinc$ structure on $\\cercle^2\\times \\mathbb{R},$ induced from the canonical one on $\\cercle^2$ and the $\\Spin$ struture on $\\mathbb{R},$ admits a parallel spinor \\cite{mo}. The restriction to $M$ of this $\\Spinc$ structure is still a $\\Spinc$ structure with a generalized Killing spinor \\cite{r2}.\\\\\\\\\nIn Section \\ref{pre}, we recall some basic facts on $\\Spinc$ structures and the restrictions of these structures to hypersurfaces. In Section \\ref{sect:3} and after giving a formula corresponding to \\eqref{eq:Fk} for any eigenspinor $\\psi$ of the square of the Dirac operator, we deduce a formula for the integral of the determinant of $T^\\psi+ Q^\\psi$ and we establish a new proof of the B\\\"{a}r-type inequality (\\ref{barspinc}). In Section \\ref{sec4}, we consider the $3$-dimensional case and treat examples of hypersurfaces in $\\CC \\PP^2$. In the last section, we come back to the question of a spinorial characterisation of surfaces in $\\cercle^2\\times \\mathbb{R}$. Here we use a different approach than the one in \\cite{r}. In fact, we prove that given any surface $M$ carrying a generalized Killing spinor associated with a particular $\\Spinc$ structure, the Energy-Momentum tensor satisfies the four compatibility equations stated by B. Daniel \\cite{daniel}. Thus there exists a local immersion of $M$ into $\\cercle^2\\times \\mathbb{R}$.\n\\section{Preliminaries}\\label{pre}\n\\setcounter{equation}{0}\nIn this section, we begin with some preliminaries concerning $\\Spinc$ structures and\nthe Dirac operator. Details can be found in \\cite{lm}, \\cite{montiel}, \\cite{fr1}, \\cite{r1} and\n\\cite{r2}.\\\\ \\\\\n{\\bf The Dirac operator on $\\Spinc$ manifolds:} Let $(M^n, g)$ be a Riemannian manifold of dimension $n\\geqslant2$ without\nboundary. We denote by ${\\rm SO}M$ the \n${\\rm SO}_n$-principal bundle over $M$ of positively oriented orthonormal frames. A\n$\\Spinc$ structure of $M$ is a $\\Spin_n^c$-principal bundle $(\\Spinc M,\\pi,M)$\n and an $\\cercle^1$-principal bundle $(\\cercle^1 M ,\\pi,M)$ together with a double\ncovering given by $\\theta: \\Spinc M \\longrightarrow {\\rm SO}M\\times_{M}\\cercle^1 M$ such\nthat $\\theta (ua) = \\theta (u)\\xi(a),$\nfor every $u \\in \\Spinc M$ and $a \\in \\Spin_n^c$, where $\\xi$ is the $2$-fold\ncovering of $\\Spin_n ^c$ over ${\\rm SO}_n\\times \\cercle^1$. \nLet $\\Sigma M := \\Spinc M \\times_{\\rho_n} \\Sigma_n$ be the associated spinor bundle\nwhere $\\Sigma_n = \\CC^{2^{[\\frac n2]}}$ and $\\rho_n : \\Spin_n^c\n\\longrightarrow \\End(\\Sigma_{n})$ denotes the complex spinor representation. A section of\n$\\Sigma M$ will be called a spinor field. The spinor bundle $\\Sigma M$ is equipped with a\nnatural Hermitian scalar product denoted by $(.,.)$. We\ndefine an $L^2$-scalar product\n$<\\psi,\\phi> = \\int_M (\\psi,\\phi) v_g,$\nfor any spinors $\\psi$ and $\\phi$. \\\\ Additionally, any connection 1-form $A: T(\\cercle^1 M)\\longrightarrow i\\RR$ on\n$\\cercle^1 M$ and the connection 1-form \n$\\omega^M$ on ${\\rm SO} M$, induce a connection\non the principal bundle ${\\rm SO} M\\times_{M} \\cercle^1 M$, and hence \na covariant derivative $\\nabla$ on $\\Gamma(\\Sigma M)$ \\cite{fr1,r2}. The curvature\nof $A$ is an imaginary valued 2-form denoted by $F_A= dA$, i.e., $F_A = i\\Omega$,\nwhere $\\Omega$ is a real valued 2-form on $\\cercle^1 M$. We know\n that $\\Omega$ can be viewed as a real valued 2-form on $M$ $\\cite{fr1, kn}$. In\nthis case $i\\Omega$ is the curvature form of the associated line bundle $L$. It is\nthe complex line bundle associated with the $\\cercle^1$-principal bundle via the\nstandard representation of the unit circle.\nFor every spinor $\\psi$, the Dirac operator is locally defined \nby $$D\\psi =\\sum_{i=1}^n e_i \\cdot \\nabla_{e_i} \\psi,$$\nwhere $(e_1,\\ldots,e_n)$ is a local oriented orthonormal tangent frame and ``$\\cdot$'' denotes the Clifford multiplication. The Dirac\noperator is an elliptic, self-adjoint operator with respect to the $L^2$-scalar\nproduct and verifies, for any spinor field $\\psi$, the Schr\\\"odinger-Lichnerowicz formula \n\\begin{eqnarray}\nD^2\\psi=\\nabla^*\\nabla\\psi+\\frac{1}{4}S\\psi+\\frac{i}{2}\\Omega\\cdot\\psi\n\\label{bochner}\n\\end{eqnarray}\nwhere $\\Omega\\cdot$ is the extension of the Clifford multiplication to differential\nforms given by \n$(e_i ^* \\wedge e_j ^*)\\cdot\\psi = e_i\\cdot e_j \\cdot\\psi$. For any spinor $\\psi \\in \\Gamma(\\Sigma M)$, we have \\cite{mo2}\n\\begin{eqnarray}\n (i\\Omega\\cdot\\psi,\\psi)\\ \\geqslant -\\frac{c_n}{2} \\vert \\Omega \\vert_g\\vert\n\\psi\\vert^2,\n\\label{cs}\n\\end{eqnarray}\nwhere $\\vert \\Omega \\vert_g$ is the norm of $\\Omega$, with respect to $g$ given by\n$\\vert \\Omega \\vert_g^2=\\sum_{i<j} (\\Omega_{ij} )^2$\nin any orthonormal local frame and $c_n = 2[\\frac\nn2]^\\frac12$. Moreover, equality holds in (\\ref{cs}) if and only\nif $\\Omega\\cdot\\psi = i \\frac {c_n}{2}\\vert\\Omega\\vert_g \\psi$.\\\\\nEvery $\\Spin$ manifold has a trivial $\\Spinc$ structure \\cite{fr1, bm}. In fact, we\nchoose the trivial line bundle with the trivial connection whose curvature $i\\Omega$\nis zero. Also every K\\\"ahler manifold $M$ of complex dimension $m$ has a canonical $\\Spinc$ structure. Let $\\ltimes$ by the K\\\"{a}hler form defined by the complex structure $J$, i.e. $\\ltimes (X, Y)= g(JX, Y)$ for all vector fields $X,Y\\in \\Gamma(TM).$ The complexified cotangent bundle \n$$T^*M\\otimes \\CC = \\Lambda^{1,0} M \\oplus \\Lambda^{0,1}M$$\ndecomposes into the $\\pm i$-eigenbundles of the complex linear extension of the complex structure. Thus, the spinor bundle of the canonical $\\Spinc$ structure is given by $$\\Sigma M = \\Lambda^{0,*} M =\\oplus_{r=0}^m \\Lambda^{0,r}M,$$\nwhere $\\Lambda^{0,r}M = \\Lambda^r(\\Lambda^{0,1}M)$ is the bundle of $r$-forms of type $(0, 1)$. The line bundle of this canonical $\\Spinc$ structure is given by $L = (K_M)^{-1}= \\Lambda^m (\\Lambda^{0,1}M)$, where $K_M$ is the canonical bundle of $M$ \\cite{fr1, bm}. This line bundle $L$ has a canonical holomorphic connection induced from the Levi-Civita connection whose curvature form is given by $i\\Omega = -i\\rho$, where $\\rho$ is the Ricci form given by $\\rho(X, Y) = \\Ric(JX, Y)$. \nWe point out that the canonical $\\Spinc$ structure on every K\\\"{a}hler manifold carries a parallel spinor \\cite{fr1, mo}.\\\\\\\\ \n{\\bf Spin$^c$ hypersurfaces and the Gauss formula:} Let $\\cZ$ be an oriented ($n+1$)-dimensional Riemannian $\\Spinc$ manifold and $M\n\\subset \\cZ$ be an oriented hypersurface. The manifold $M$ inherits a $\\Spinc$\nstructure induced from the one on $\\cZ$, and we have \\cite{r2}\n$$ \\Sigma M\\simeq \\left\\{\n\\begin{array}{l}\n\\Sigma \\cZ_{|_M} \\ \\ \\ \\ \\ \\ \\text{\\ \\ \\ if\\ $n$ is even,} \\\\\\\\\n \\SZp_{|_M} \\ \\text{\\ \\ \\ \\ \\ \\ if\\ $n$ is odd.}\n\\end{array}\n\\right.\n$$\nMoreover Clifford multiplication by a vector field $X$, tangent to $M$, is given by \n\\begin{eqnarray}\nX\\bullet\\phi = (X\\cdot\\nu\\cdot \\psi)_{|_M},\n\\label{Clifford}\n\\end{eqnarray}\nwhere $\\psi \\in \\Gamma(\\Sigma \\cZ)$ (or $\\psi \\in \\Gamma(\\SZp)$ if $n$ is odd),\n$\\phi$ is the restriction of $\\psi$ to $M$, ``$\\cdot$'' is the Clifford\nmultiplication on $\\cZ$, ``$\\bullet$'' that on $M$ and $\\nu$ is the unit normal\nvector. The connection 1-form defined on the restricted $\\SSS^1$-principal bundle $(\\PSM :=\n\\PSZ_{|_M},\\pi,M)$, is given by $A= {A^\\cZ}_{|_M} : T(\\PSM) = T(\\PSZ)_{|_M}\n\\longrightarrow i\\RR.$ Then the curvature 2-form $i\\Omega$ on the\n$\\SSS^1$-principal bundle $\\PSM$ is given by $i\\Omega= {i\\Omega^\\cZ}_{|_M}$,\nwhich can be viewed as an imaginary 2-form on $M$ and hence as the curvature form of\nthe line bundle $L^M$, the restriction of the line bundle $L^\\cZ$ to $M$. For every\n$\\psi \\in \\Gamma(\\SZ)$ ($\\psi \\in \\Gamma(\\SZp)$ if $n$ is odd), the real 2-forms\n$\\Omega$ and $\\Omega^\\cZ$ are related by \\cite{r2}\n\\begin{eqnarray}\n(\\Omega^\\cZ \\cdot\\psi)_{|_M} = \\Omega\\bullet\\phi -\n(\\nu\\lrcorner\\Omega^\\cZ)\\bullet\\phi.\n\\label{glucose}\n\\end{eqnarray}\nWe denote by $\\nabla^{\\Sigma \\cZ}$ the spinorial Levi-Civita connection on $\\Sigma\n\\cZ$ and by $\\nabla$ that on $\\Sigma M$. For all $X\\in \\Gamma(TM)$, we have the spinorial Gauss formula \\cite{r2}:\n\\begin{equation}\n(\\nabla^{\\Sigma \\cZ}_X\\psi)_{|_M} = \\nabla_X \\phi + \\half II(X)\\bullet\\phi,\n\\label{spingauss}\n\\end{equation}\nwhere $II$ denotes the Weingarten map of the hypersurface. Moreover, Let $D^\\cZ$ and $D^M$ be the Dirac operators on $\\cZ$ and $M$, after\ndenoting by the same symbol any spinor and its restriction to $M$, we have\n\\begin{equation}\n\\nu\\cdot D^\\cZ\\phi = \\DDM\\phi +\\frac{n}{2}H\\phi -\\nabla^{\\Sigma\\cZ}_{\\nu}\\phi,\n\\label{diracgauss}\n\\end{equation}\nwhere $H = \\frac1n \\tr(II)$ denotes the mean curvature and $\\DDM = D^M$ if $n$ is even and $\\DDM=D^M \\oplus(-D^M)$ if $n$ is odd.\n\\section{The $2$-dimensional case}\\label{sect:3}\n\\setcounter{equation}{0}\nIn this section, we consider compact surfaces endowed with any $\\Spinc$ structure. We have \n\\begin{thm}\\label{procarrdir}\n Let $(M^2, g)$ be a Riemannian manifold and $\\psi$ an eigenspinor of the\nsquare of the Dirac operator $D^2$ with eigenvalue $\\lambda^2$ associated with any $\\Spinc$ structure. Then we have \n$$ \\lambda^2 = \\frac{S}{4} + {\\vert{T}^{\\psi} \\vert^2} + \\vert{Q}^{\\psi} \\vert^2 +\\Delta f+|Y|^2-2Y(f)+(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}), $$ \nwhere $f$ is the real-valued function defined by $f=\\frac{1}{2}{\\rm ln}|\\psi|^2$\nand $Y$ is a vector field on $TM$ given by $g(Y,Z)=\n\\frac{1}{\\vert\\psi\\vert^2} \\Re(D\\psi, Z\\cdot\\psi)$ for any $Z \\in \\Gamma(TM)$.\n\\label{thm1}\n\\end{thm}\n{\\bf Proof.} Let $\\{e_1, e_2\\}$ be an orthonormal frame of $TM$. Since the spinor bundle $\\Sigma M$ is of real dimension $4$, the set $\\{\\frac{\\psi}{|\\psi|}, \\frac{e_1\\cdot\\psi}{|\\psi|},\\frac{e_2\\cdot\\psi}{|\\psi|},\\frac{e_1\\cdot e_2\\cdot\\psi}{|\\psi|}\\}$ is orthonormal with respect to the real product $\\Re(\\cdot, \\cdot)$. The covariant derivative of $\\psi$ can be expressed in this frame as \n\\begin{eqnarray} \n \\nabla_X\\psi = \\delta(X) \\psi + \\alpha(X)\\cdot\\psi + \\beta(X) e_1\\cdot\ne_2\\cdot\\psi,\n\\label{equa1}\n\\end{eqnarray}\nfor all vector fields $X,$ where $\\delta$ and $\\beta$ are $1$-forms and $\\alpha$ is a $(1,1)$-tensor field. Moreover $\\beta$, $\\delta$ and $\\alpha$ are uniquely\ndetermined by the spinor $\\psi$. In fact, taking the scalar product of (\\ref{equa1}) respectively with $\\psi, e_1\\cdot\\psi, e_2\\cdot\\psi, e_1\\cdot e_2\\cdot\\psi$, we get $\\delta =\\frac{d(\\vert\\psi\\vert^2)}{2\\vert\\psi\\vert^2}$ and\n$$\\alpha (X)=- \\ell^\\psi(X)+q^\\psi(X)\\quad\\text{and}\\quad\\beta(X)=\\frac{1}{\\vert\\psi\\vert^2} \\Re(\\nabla_X\\psi, e_1\\cdot e_2\\cdot\\psi).$$ \nUsing (\\ref{bochner}), it follows that\n$$\\lambda^2 = \\frac {\\Delta ( \\vert \\psi \\vert^2 )} { 2 \\vert \\psi \\vert^2 } +\n\\vert \\alpha \\vert^2 + \\vert \\beta\\vert^2 + \\vert \\delta\\vert^2 +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).$$\nNow it remains to compute the term $\\vert\\beta\\vert^2$. We have\n\\begin{eqnarray*}\n\\vert\\beta\\vert^2 &=& \\frac{1}{\\vert\\psi\\vert^4}\\Re(\\nabla_{e_1}\\psi, e_1\\cdot e_2\n\\cdot\\psi)^2+\\frac{1}{\\vert\\psi\\vert^4}\\Re(\\nabla_{e_2}\\psi, e_1\\cdot e_2\n\\cdot\\psi)^2\\\\\n&=& \\frac{1}{\\vert\\psi\\vert^4} \\Re(D\\psi - e_2 \\cdot\\nabla_{e_2}\\psi, e_2 \\cdot\\psi)^2\n+ \\frac{1}{\\vert\\psi\\vert^4} \\Re(D\\psi - e_1 \\cdot\\nabla_{e_1}\\psi, e_1 \\cdot\\psi)^2 \\\\\n&=& g(Y, e_1)^2 + g(Y, e_2)^2 + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\n\\vert\\psi\\vert^4} - g(Y, \\frac{d(\\vert\\psi\\vert^2)}{\\vert\\psi\\vert^2})\\\\\n&=& |Y|^2-2Y(f) + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\n\\vert\\psi\\vert^4},\n\\end{eqnarray*}\nwhich gives the result by using the fact that $\\Delta f=\\frac {\\Delta ( \\vert \\psi \\vert^2)} {2\\vert \\psi \\vert^2}+\\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{2\n\\vert\\psi\\vert^4}.$\n\\hfill$\\square$\n\\begin{remark} \\label{rempropre} Under the same conditions as Theorem \\ref{thm1}, if $\\psi$ is an eigenspinor of $D$ with eigenvalue $\\lambda$, we get\n$$ \\lambda^2 = \\frac{S}{4} + {\\vert{T}^{\\psi} \\vert^2} +\\Delta f+(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).$$ \nIn fact, in this case $Y=0$ and \n\\begin{eqnarray}\n0 &=& \\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)=\\Re(e_1\\cdot\\nabla_{e_1}\\psi+e_2\\cdot\\nabla_{e_2}\\psi,e_1\\cdot e_2\\cdot\\psi)\\nonumber \\\\\n&=&\\Re(-e_2\\cdot\\nabla_{e_1}\\psi+e_1\\cdot\\nabla_{e_2}\\psi,\\psi) =2Q^\\psi(e_1,e_2)|\\psi|^2.\n\\label{qpsi}\n\\end{eqnarray} \nThis was proven by Friedrich and Kim in \\cite{fk} for a $\\Spin$ structure on $M$.\n\\end{remark}\nIn the following, we will give an estimate for the integral $\\displaystyle\\int_M {\\rm det}(T^\\psi+Q^\\psi) v_g$ in terms of geometric quantities, which has the advantage that it does not depend on the eigenvalue $\\lambda$ nor on the eigenspinor $\\psi$. This is a generalization of the result of Friedrich and Kim in \\cite{fk} for $\\Spin$ structures.\n\\begin{thm} \nLet $M$ be a compact surface and $\\psi$ any eigenspinor of $D^2$ associated with eigenvalue $\\lambda^2$. Then we have \n\\begin{eqnarray}\n\\int_M {\\rm det}(T^\\psi+Q^\\psi)v_g \\geq \\frac{\\pi\\chi(M)}{2}- \\frac 14 \\int_M \\vert\\Omega\\vert v_g.\n\\label{dett}\n\\end{eqnarray}\nEquality in \\eqref{dett} holds if and only if either $\\Omega$ is zero or has constant sign. \n\\label{D2}\n\\end{thm} \n{\\bf Proof.} As in the previous theorem, the spinor $D\\psi$ can be expressed in the orthonormal frame of the spinor bundle. Thus the norm of $D\\psi$ is equal to\n\\begin{eqnarray}\n|D\\psi|^2&=&\\frac{1}{|\\psi|^2}\\Re(D\\psi,\\psi)^2+\\frac{1}{|\\psi|^2}\\sum_{i=1}^2 \\Re(D\\psi,e_i\\cdot\\psi)^2+\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2\\nonumber\\\\\n&=&({\\tr}\\,T^\\psi)^2|\\psi|^2+|Y|^2|\\psi|^2+\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2,\n\\label{eq:Dirac}\n\\end{eqnarray}\nwhere we recall that the trace of $T^\\psi$ is equal to $-\\frac{1}{|\\psi|^2}\\Re(D\\psi,\\psi).$ On the other hand, by (\\ref{qpsi}) we have that\n$\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2=2|Q^\\psi|^2|\\psi|^2.$ Thus Equation \\eqref{eq:Dirac} reduces to \n$$\\frac{|D\\psi|^2}{|\\psi|^2}=({\\tr}\\, T^\\psi)^2+|Y|^2+2|Q^\\psi|^2.$$\nNow with the use of the equality $\\Re(D^2\\psi,\\psi)=|D\\psi|^2-{\\rm div}\\xi,$ where $\\xi$ is the vector field given by $\\xi=|\\psi|^2Y,$ we get \n\\begin{equation}\n\\lambda^2+\\frac{1}{|\\psi|^2}{\\rm div}\\xi=({\\tr}\\, T^\\psi)^2+|Y|^2+2|Q^\\psi|^2.\n\\label{eq:Dirac2}\n\\end{equation} \nAn easy computation leads to $\\frac{1}{|\\psi|^2}{\\rm div}\\xi={\\rm div} Y+2Y(f)$ where we recall that $f=\\frac{1}{2}{\\rm ln}(|\\psi|^2).$ Hence substituting this formula into \\eqref{eq:Dirac2} and using Theorem \\ref{thm1} yields \n$$\\frac{S}{4} + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}) + \\Delta f+{\\rm div} Y= ({\\tr} T^\\psi)^2+|Q^\\psi|^2-|T^\\psi|^2=2{\\rm det}(T^\\psi+Q^\\psi).$$\nFinally integrating over $M$ and using the Gauss-Bonnet formula, we deduce the required result with the help of Equation (\\ref{cs}). Equality holds if and only if $\\Omega\\cdot\\psi = i\\vert\\Omega\\vert \\psi$. In the orthonormal frame $\\{e_1, e_2\\}$, the $2$-form $\\Omega$ can be written $\\Omega= \\Omega_{12} \\ e_1 \\wedge e_2$, where $\\Omega_{12}$ is a function defined on $M$. Using the decomposition of $\\psi$ into positive and negative spinors $\\psi^+$ and $\\psi^-$, we find that the equality is attained if and only if\n$$\\Omega_{12} \\ e_1 \\cdot e_2 \\cdot\\psi^+ + \\Omega_{12}\\ e_1 \\cdot e_2 \\cdot\\psi^- = i \\vert\\Omega_{12} \\vert \\psi^+ +i \\vert\\Omega_{12} \\vert \\psi^-,$$\nwhich is equivalent to say that,\n$$\\Omega_{12} \\psi^+ = -\\vert\\Omega_{12}\\vert \\psi^+\\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\Omega_{12} \\psi^- = \\vert\\Omega_{12}\\vert \\psi^-.$$\nNow if $\\psi^+ \\neq 0$ and $\\psi^- \\neq 0$, we get $\\Omega = 0$. Otherwise, it has constant sign. In the last case, we get that $\\int_M|\\Omega|v_g=2\\pi\\chi(M),$ which means that the l.h.s. of this equality is a topological invariant.\n\\hfill$\\square$\\\\ \\\\\n\nNext, we will give another proof of the B\\\"ar-type inequality (\\ref{barspinc}) for the eigenvalues of any $\\Spinc$ Dirac operator. The following theorem was proved by the second author in \\cite{r1} using conformal deformation of the spinorial Levi-Civita connection. \n\\begin{thm}\n Let $M$ be a compact surface. For any $\\Spinc$ structure on $M$, any eigenvalue $\\lambda$ of the Dirac operator $D$ to which is attached an eigenspinor $\\psi$ satisfies\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\frac{2\\pi \\chi(M)}{Area(M)} - \\frac{1}{Area(M)} \\int_M \\vert\\Omega\\vert v_g.\n\\label{inequalitybar}\n\\end{eqnarray}\nEquality holds if and only if the eigenspinor $\\psi$ is a\n $\\Spinc$ Killing spinor, i.e., it satisfies $\\Omega\\cdot\\psi = i \\vert\\Omega\\vert\\psi$ and $\\nabla_X \\psi = -\\frac{\\lambda}{2} X\\cdot\\psi$ for any $X \\in \\Gamma(TM)$.\n\\label{thmspinc}\n\\end{thm}\n{\\bf Proof.} With the help of Remark \\eqref{rempropre}, we have that\n\\begin{eqnarray}\n \\lambda^2 = \\frac{S}{4} + \\vert T^\\psi\\vert^2 + \\bigtriangleup f +(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).\n\\label{barspinccc}\n\\end{eqnarray}\nSubstituting the Cauchy-Schwarz inequality, i.e. $\\vert T^\\psi\\vert^2 \\geqslant \\frac{\\lambda^2}{2}$ and the estimate (\\ref{cs}) into Equality (\\ref{barspinccc}), we easily deduce the result after integrating over $M$. Now the equality in \\eqref{inequalitybar} holds if and only if the eigenspinor $\\psi$ satisfies $\\Omega\\cdot\\psi = i \\vert\\Omega\\vert\\psi$ and $\\vert T^\\psi\\vert^2= \\frac{\\lambda^2}{2}$. Thus, the second equality is equivalent to say that $\\ell^\\psi (X) = \\frac{\\lambda}{2}X$ for all $X\\in \\Gamma(TM)$. \nFinally, a straightforward computation of the spinorial curvature of the spinor field $\\psi$ gives in a local frame $\\{e_1,e_2\\}$ after using the fact $\\beta=-(*\\delta)$ that\n\\begin{eqnarray*}\n\\frac{1}{2}R_{1212}\\ e_1\\cdot e_2\\cdot\\psi&=&\\Big(\\frac{\\lambda^2}{2}+e_1(\\delta(e_1))+e_2(\\delta(e_2))\\Big)e_2\\cdot e_1\\cdot\\psi-\\lambda\\delta(e_2) e_1\\cdot\\psi\\\\&&+\\lambda \\delta(e_1)e_2\\cdot\\psi+\\Big(e_1(\\delta(e_2))-e_2(\\delta(e_1))\\Big)\\psi.\n\\end{eqnarray*}\nThus the scalar product with $e_1\\cdot\\psi$ and $e_2\\cdot\\psi$ implies that $\\delta=0$. Finally, $\\beta=0$ and the eigenspinor $\\psi$ is a $\\Spinc$ Killing spinor.\n\\hfill$\\square$\\\\ \\\\\nNow, we will give some examples where equality holds in (\\ref{inequalitybar}) or in \\eqref{dett}. Some applications of Theorem \\ref{thm1} are also given.\\\\ \\\\\n{\\bf Examples:}\n\\begin{enumerate}\n \\item Let $\\cercle^2$ be the round sphere equipped with the standard metric of curvature one. As a K\\\"ahler manifold, we endow the sphere with the canonical $\\Spinc$ structure of curvature form equal to $i\\Omega=-i\\ltimes$, where $\\ltimes$ is the K\\\"{a}hler $2$-form. Hence, we have $\\vert\\Omega\\vert = \\vert\\ltimes\\vert =1$. Furthermore, we mentionned that for the canonical $\\Spinc$ structure, the sphere carries parallel spinors, i.e., an eigenspinor associated with the eigenvalue $0$ of the Dirac operator $D$. Thus equality holds in (\\ref{inequalitybar}). On the other hand, the equality in \\eqref{dett} also holds, since the sign of the curvature form $\\Omega$ is constant. \n\\item Let $f: M \\rightarrow \\cercle^3$ be an isometric immersion of a surface\n$M^2$ into the sphere equipped with its unique $\\Spin$ structure and assume that the mean curvature $H$ is\nconstant. The restriction of a Killing spinor on $\\cercle^3$ to the surface $M$ defines a spinor field\n$\\phi$ solution of the following equation \\cite{gj}\n\\begin{eqnarray}\n \\nabla_X \\phi = -\\frac 12 II(X) \\bullet \\phi + \\frac 12 J(X)\\bullet\\phi,\n\\label{generalized}\n\\end{eqnarray}\nwhere $II$ denotes the second fundamental form of the surface and $J$ is the complex\nstructure of $M$ given by\nthe rotation of angle $\\frac{\\pi}{2}$ on $TM$. It is\neasy to check that $\\phi$ is an eigenspinor for $D^2$ associated with the\neigenvalue $H^2 +1$. Moreover $D \\phi = H \\phi + e_1\\cdot e_2\\cdot \\phi$, so that $Y=0$. Moreover the tensor $T^\\phi=\\frac{1}{2}II$ and $Q^\\phi=\\frac{1}{2}J$. Hence by Theorem \\ref{thm1}, and since the norm of $\\phi$ is\nconstant, we obtain\n$$H^2 + \\frac12 = \\frac14 S +\\frac{1}{4}|II|^2.$$\n\\item On two-dimensional manifolds, we can define another Dirac operator associated with the complex\nstructure $J$ given by $\\widetilde D =Je_1 \\cdot \\nabla_{e_1} +Je_2 \\cdot \\nabla_{e_2} = e_2\\cdot \\nabla_{e_1}- e_1 \\cdot\\nabla_{e_2}$. Since $\\widetilde D$ satisfies $D^2 = (\\widetilde D)^2$, all the above results are also true for the eigenvalues of $\\widetilde D$.\n\n\\item Let $M^2$ be a surface immersed in $\\cercle^2 \\times \\RR$. The product of the canonical $\\Spinc$ structure on $\\cercle^2$ and the unique $\\Spin$ structure on $\\RR$ define a $\\Spinc$ structure on $\\cercle^2\\times \\RR$ carrying parallel spinors \\cite {mo}. Moreover, by the Schr\\\"{o}dinger-Lichnerowicz formula, any parallel spinor $\\psi$ satisfies $\\Omega^{\\cercle^2\\times \\mathbb{R}}\\cdot\\psi=i\\psi$, where $\\Omega^{\\cercle^2\\times \\mathbb{R}}$ is the curvature form of the auxiliary line bundle. Let $\\nu$ be a unit normal vector field of the surface. We then write $\\partial t=T+f\\nu$ where $T$ is a vector field on $TM$ with $||T||^2+f^2=1$. On the other hand, the vector field $T$ splits into $T=\\nu_1+h\\partial t$ where $\\nu_1$ is a vector field on the sphere. The scalar product of the first equation by $T$ and the second one by $\\partial t$ gives $||T||^2=h$ which means that $h=1-f^2$. Hence the normal vector field $\\nu$ can be written as $\\nu=f\\partial t-\\frac{1}{f}\\nu_1.$ As we mentionned before, the $\\Spinc$ structure on $\\cercle^2 \\times \\RR$ induces a $\\Spinc$ structure on $M$ with induced auxiliary line bundle. Next, we want to prove that the curvature form of the auxiliary line bundle of $M$ is equal to $i\\Omega(e_1,e_2)=-if$, where $\\{e_1,e_2\\}$ denotes a local orthonormal frame on $TM$. Since the spinor $\\psi$ is parallel, we have by \\cite{mo} that for all $X\\in T(\\cercle^2 \\times \\RR)$ the equality ${\\rm Ric}^{\\cercle^2 \\times \\RR} X\\cdot\\psi=i(X\\lrcorner\\Omega^{\\cercle^2 \\times \\RR})\\cdot\\psi$. Therefore, we compute\n\\begin{eqnarray*}\n(\\nu\\lrcorner \\Omega^{\\cercle^2 \\times \\RR})\\bullet\\phi&=&(\\nu\\lrcorner \\Omega^{\\cercle^2 \\times \\RR})\\cdot\\nu\\cdot\\psi|_M=i\\nu\\cdot{\\rm Ric}^{\\cercle^2 \\times \\RR}\\ \\nu\\cdot\\psi|_M\\\\\n&=&-\\frac{1}{f}i\\nu\\cdot\\nu_1\\cdot\\psi|_M=i\\nu\\cdot (\\nu-f\\partial t).\\psi|_M\\\\\n&=&(-i\\psi-if\\nu\\cdot\\partial t\\cdot\\psi)|_M. \n\\end{eqnarray*}\nHence by Equation \\eqref{glucose}, we get that $\\Omega\\bullet\\phi=-i (f\\nu\\cdot\\partial t\\cdot\\psi)|_M.$ The scalar product of the last equality with $e_1\\cdot e_2\\cdot\\psi$ gives\n$$\\Omega(e_1,e_2)|\\phi|^2=-f\\Re(i\\nu\\cdot\\partial t\\cdot\\psi,e_1\\cdot e_2\\cdot\\psi)|_M= -f\\Re(i\\partial t\\cdot\\psi,\\psi)|_M.$$\nWe now compute the term $i\\partial t\\cdot\\psi$. For this, let $\\{e'_1, J e'_1\\}$ be a local orthonormal frame of the sphere $\\cercle^2$. The complex volume form acts as the identity on the spinor bundle of $\\cercle^2 \\times \\RR$, hence $\\partial t \\cdot\\psi = e'_1\\cdot Je'_1\\cdot\\psi$. But we have\n$$\\Omega^{\\cercle^2 \\times \\RR}\\cdot\\psi = -\\rho\\cdot\\psi = -\\ltimes\\cdot\\psi = -e'_1\\cdot Je'_1\\cdot\\psi.$$\nTherefore, $i\\partial t\\cdot\\psi = \\psi$. Thus we get $\\Omega(e_1,e_2)=-f$. Finally, \n$$(i\\Omega\\bullet\\phi,\\phi) = f \\Re(\\nu\\cdot\\partial t\\cdot\\psi,\\psi)|_M=-fg(\\nu,\\partial t)\\vert\\phi\\vert^2=-f^2\\vert\\phi\\vert^2.$$ Hence Equality in Theorem \\ref{thm1} is just \n$$H^2=\\frac{S}{4}+\\frac{1}{4}|II|^2-\\frac{1}{2}f^2.$$\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\\section{The 3-dimensional case}\\label{sec4} \n\\setcounter{equation}{0}\nIn this section, we will treat the $3$-dimensional case.\n\\begin{thm}\n Let $(M^3,g)$ be an oriented Riemannian manifold. For any $\\Spinc$ structure on $M$, any eigenvalue\n$\\lambda$ of the Dirac operator to which is attached an eigenspinor $\\psi$ satisfies\n$$\\lambda^2 \\leqslant \\frac{1}{\\vol(M, g)} \\int_M (\\vert T^\\psi\\vert^2 +\\frac S4\n+\\frac{\\vert\\Omega\\vert}{2})v_g.$$\nEquality holds if and only if the norm of $\\psi$ is constant and $\\Omega\\cdot\\psi\n= i\\vert\\Omega\\vert \\psi$. \n\\label{thm4}\n\\end{thm}\n\n{\\bf Proof.} As in the proof of Theorem \\ref{thm1}, the set $\\{\\frac{\\psi}{|\\psi|}, \\frac{e_1\\cdot\\psi}{|\\psi|},\\frac{e_2\\cdot\\psi}{|\\psi|},\\frac{e_3\\cdot\\psi}{|\\psi|}\\}$ is orthonormal with respect to the real product $\\Re(\\cdot, \\cdot)$. The covariant derivative of $\\psi$ can be expressed in this frame as \n\\begin{eqnarray} \n \\nabla_X\\psi = \\eta(X) \\psi + \\ell(X)\\cdot\\psi,\n\\label{eq1}\n\\end{eqnarray}\nfor all vector fields $X,$ where $\\eta$ is a $1$-form and $\\ell$ is a $(1,1)$-tensor field. Moreover $\\eta =\\frac{d(\\vert\\psi\\vert^2)}{2\\vert\\psi\\vert^2}$ and $\\ell(X)=- \\ell^\\psi(X)$. Using (\\ref{bochner}), it follows that\n\\begin{eqnarray*}\n\\lambda^2 &=& \\frac {\\Delta ( \\vert \\psi \\vert^2 )} { 2 \\vert \\psi \\vert^2 } +\n\\vert T^\\psi \\vert^2 + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\\vert\\psi\\vert^4} +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2})\\\\\n&=& \\Delta f - \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{2\\vert\\psi\\vert^4} \n+\\vert T^\\psi \\vert^2 +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).\n \\end{eqnarray*}\nBy the Cauchy-Schwarz inequality, we have $\\frac 12 (i\\Omega \\cdot\\psi,\\frac{\\psi}{\\vert\\psi\\vert^2}) \\leqslant \\frac 12 \\vert\\Omega\\vert$. Integrating over $M$ and using the fact that $\\vert d(\\vert\\psi\\vert^2)\\vert^2\\geqslant 0$, we get the result.\n\\hfill$\\square$\n\n\n\n\\begin{example}\nLet $M^3$ be a 3-dimensional Riemannian manifold immersed in $\\CC \\PP^2$ with\nconstant mean curvature $H$. Since $\\CC \\PP^2$ is a K\\\"{a}hler manifold, we endow it\nwith the canonical $\\Spinc$ structure whose line bundle has\ncurvature equal to $-3i\\ltimes$. Moreover, by the Schr\\\"{o}dinger-Lichnerowicz formula we have that any parallel spinor $\\psi$ satisfies $\\Omega^{\\CC \\PP^2} \\cdot\\psi = 6i \\psi.$ As in the previous example, we compute \n $$(\\nu\\lrcorner\\Omega^{\\CC \\PP^2})\\bullet\\phi = i (\\nu\\cdot \\Ric^{\\CC\n\\PP^2}(\\nu)\\cdot\\psi)_{\\mid_M} = -3i \\phi.$$ \nFinally, $\\Omega\\bullet\\phi = 3i\\phi.$ Using Equation (\\ref{diracgauss}), we have that $-\\frac 32 H$ is an eigenvalue of $D$. Since the norm of $\\phi$ is constant, equality holds in Theorem \\ref{thm4} and hence\n$$\\frac 94 H^2 + \\frac 32 = \\frac{S}{4}+\\frac{1}{4}\\vert II\\vert^2.$$\n\\end{example}\n\n\\section{Characterization of surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$}\\label{charac} \n\\setcounter{equation}{0}\nIn this section, we characterize the surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$ by solutions of the generalized Killing spinors equation which are restrictions of parallel spinors of the canonical $\\Spinc$-structure on $\\mathbb{S}^2\\times \\mathbb{R}$ (see also \\cite{r} for a different proof). First recall the compatibility equations for characterization of surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$ established by B. Daniel \\cite[Thm 3.3]{daniel}: \n\\begin{thm} \\label{daniel}Let $(M,g)$ be a simply connected Riemannian manifold of dimension $2$, $A:TM\\rightarrow TM$ a field of symmetric operator and $T$ a vector field on $TM$. We denote by $f$ a real valued function such that $f^2+||T||^2=1$. Assume that $M$ satisfies the Gauss-Codazzi equations, i.e. $G={\\rm det} A+f^2$ and\n$$ d^\\nabla A(X,Y):=(\\nabla_XA)Y-(\\nabla_YA)X=f(g(Y,T)X-g(X,T)Y),$$ \nwhere $G$ is the gaussian curvature, and the following equations \n$$\\nabla_X T=fA(X), \\,\\,\\, X(f)=-g(AX,T).$$ Then there exists an isometric immersion of $M$ into $\\mathbb{S}^2\\times \\mathbb{R}$ such that the Weingarten operator is $A$ and $\\partial t=T+f\\nu,$ where $\\nu$ is the normal vector field to the surface $M$. \n\\end{thm}\nNow using this characterization theorem, we state our result:\n\\begin{thm} \\label{carac}\nLet $M$ be an oriented simply connected Riemannian manifold of dimension $2$. Let $T$ be a vector field and denote by $f$ a real valued function such that $f^2+||T||^2=1$. Denote by $A$ a symmetric endomorphism field of $TM$. The following statements are equivalent: \n\\begin{enumerate}\n\\item There exists an isometric immersion of $M$ into $\\mathbb{S}^2\\times \\mathbb{R}$ of Weingarten operator $A$ such that $\\partial t=T+f\\nu,$ where $\\nu$ is the unit normal vector field of the surface.\n\\item There exists a $\\Spinc$ structure on $M$ whose line bundle has a connection of curvature given by $i\\Omega=-if \\ltimes,$ such that it carries a non-trivial solution $\\phi$ of the generalized Killing spinor equation $\\nabla_X\\phi=-\\frac{1}{2}AX\\bullet\\phi$, with $T\\bullet\\phi=-f\\phi+\\bar\\phi.$\n\\end{enumerate}\n\\end{thm}\n{\\bf Proof.} We begin with $1\\Rightarrow 2$. The existence of such a $\\Spinc$ structure is assured by the restriction of the canonical one on $\\cercle^2\\times \\mathbb{R}$. Moreover, using the spinorial Gauss formula \\eqref{spingauss}, any parallel spinor $\\psi$ on $\\cercle^2\\times \\mathbb{R}$ induces a generalized Killing spinor $\\phi=\\psi|_M$ with $A$ the Weingarten map of the surface $M$. Hence it remains to show the relation $T\\bullet\\phi=-f\\phi+\\bar\\phi$. In fact, using that $\\Omega^{\\cercle^2\\times \\mathbb{R}}\\cdot\\psi=i\\psi,$ we write in the frame $\\{e_1,e_2,\\nu\\}$ \n\\begin{equation}\\label{eq:3}\n\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)e_1\\cdot e_2\\cdot\\psi+\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\nu)e_1\\cdot\\nu\\cdot\\psi+\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_2,\\nu)e_2\\cdot\\nu\\cdot\\psi=i\\psi.\n\\end{equation}\nBy the previous example in Section \\ref{sect:3}, we know that $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)=-f$. For the other terms, we compute \n$$\n\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\nu)=\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\frac{1}{f}\\partial t-\\frac{1}{f}T)=-\\frac{1}{f}g(T,e_2)\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)=g(T,e_2),\n$$\nwhere the term $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\partial t)$ vanishes since we can split $e_1$ into a sum of vectors on the sphere and on $\\mathbb{R}.$ Similarly, we find that $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_2,\\nu)=-g(T,e_1).$ By substituting these values into \\eqref{eq:3} and taking Clifford multiplication with $e_1\\cdot e_2$, we get the desired property. For $2\\Rightarrow 1$, a straightforward computation for the spinorial curvature of the generalized Killing spinor $\\phi$ yields on a local frame $\\{e_1,e_2\\}$ of $TM$ that \n\\begin{equation}\n(-G+{\\rm det}\\,A)e_1\\bullet e_2\\bullet\\phi=-(d^\\nabla A)(e_1,e_2)\\bullet \\phi + if\\phi.\n\\label{eq:6}\n\\end{equation}\nIn the following, we will prove that the spinor field $\\theta:=i\\phi-if\\bar{\\phi}+JT\\bullet\\phi$ is zero. For this, it is sufficient to prove that its norm vanishes. Indeed, we compute \n\\begin{equation}\n|\\theta|^2=|\\phi|^2+f^2|\\phi|^2+||T||^2|\\phi|^2-2\\Re(i\\phi,if\\bar{\\phi})+2\\Re(i\\phi,JT\\bullet\\phi)\n\\label{eq:5}\n\\end{equation}\nFrom the relation $T\\bullet\\phi=-f\\phi+\\bar{\\phi}$ we deduce that $\\Re(\\phi,\\bar\\phi)=f|\\phi|^2$ and the equalities\n$$g(T,e_1)|\\phi|^2=\\Re(ie_2\\bullet\\phi,\\phi) \\quad\\text{and}\\quad g(T,e_2)|\\phi|^2=-\\Re(ie_1\\bullet\\phi,\\phi).$$\nTherefore, Equation \\eqref{eq:5} becomes \n\\begin{eqnarray*}\n|\\theta|^2&=&2|\\phi|^2-2f^2|\\phi|^2+2\\Re(i\\phi,JT\\bullet\\phi)\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2+2 g(JT,e_1)\\Re(i\\phi,e_1\\bullet\\phi)+2g(JT,e_2) \\Re(i\\phi,e_2\\bullet\\phi)\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2+2 g(JT,e_1) g(T,e_2)|\\phi|^2-2g(JT,e_2) g(T,e_1)|\\phi|^2\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2-2g(JT,e_1)^2|\\phi|^2-2g(T,e_1)^2|\\phi|^2\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2-2||T||^2|\\phi|^2=0.\n\\end{eqnarray*}\nThus, we deduce $if\\varphi=-f^2e_1\\cdot e_2\\cdot\\varphi-fJT\\cdot\\varphi$, where we use the fact that $\\bar\\phi=i e_1\\bullet e_2\\bullet\\phi$. In this case, Equation \\eqref{eq:6} can be written as \n$$(-G+{\\rm det}\\,A+f^2)e_1\\bullet e_2\\bullet\\phi=-((d^\\nabla A)(e_1,e_2)+fJT)\\bullet \\phi.$$\nThis is equivalent to say that both terms $R_{1212}+{\\rm det}\\,A+f^2$ and $(d^\\nabla A)(e_1,e_2)+fJT$ are equal to zero. In fact, these are the Gauss-Codazzi equations in Theorem \\ref{daniel}. In order to obtain the two other equations, we simply compute the derivative of $T\\cdot\\varphi=-f\\varphi+\\bar\\varphi$ in the direction of $X$ to get \n\\begin{eqnarray*}\n\\nabla_X T \\bullet \\phi+T\\bullet\\nabla_X\\varphi&=&\\nabla_X T \\bullet \\phi-\\frac{1}{2}T\\bullet A(X)\\bullet\\varphi\\\\\n&=&-X(f)\\varphi-f\\nabla_X\\varphi+\\nabla_X\\bar\\varphi\\\\\n&=&-X(f)\\varphi+\\frac{1}{2}f AX\\bullet\\varphi+\\frac{1}{2} AX\\bullet\\bar{\\varphi}\\\\\n&=&-X(f)\\varphi+\\frac{1}{2}f AX\\bullet\\varphi+\\frac{1}{2} AX\\bullet(T\\bullet\\varphi+f\\varphi).\n\\end{eqnarray*}\nThis reduces to $\\nabla_X T \\bullet \\phi+g(T,A(X))\\varphi=-X(f)\\varphi+fA(X)\\bullet\\varphi.$ Hence we obtain $X(f)=-g(A(X),T)$ and $\\nabla_X T=fA(X)$ which finishes the proof.\n\\hfill$\\square$\n\\begin{remark} The second condition in Theorem \\ref{carac} is equivalent to the existence of a $\\Spinc$ structure whose line bundle $L$ verifies \n$c_1(L)=[\\frac{i}{2\\pi}f\\ltimes]$ and $f \\ltimes$ is a closed 2-form. This $\\Spinc$ structure carries a non-trivial solution $\\phi$ of the generalized Killing spinor equation $\\nabla_X\\phi=-\\frac{1}{2}AX\\bullet\\phi$, with $T\\bullet\\phi=-f\\phi+\\bar\\phi.$\n\\end{remark}\n{\\bf Acknowledgment}\\\\\\\\\nBoth authors are grateful to Oussama Hijazi for his encouragements and relevant remarks.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} |
