In Chapter One we have defined characteristic classes and numbers. A natural question is hoe to compute these characteristic numbers. Let $\\omega$ be a characteristic form on an even dimensional smooth closed oriented manifold $M$. If
\n\\[
\n\\omega=\\omega_{[1]}+\\omega_{[2]}+\\cdots+\\omega_{[\\dim M]},\\quad \\omega_{[i]}\\in\\Omega^{i}(M), i=1,\\cdots,\\dim M,
\n\\] then the characteristic number associated $\\omega$ is defined by $\\int_{M}\\omega=\\int_M\\omega_{[\\dim M]}$. The Bott’s result shows that
\n\\[
\n\\int_M\\omega_{[\\dim M]}=\\sum_{p\\in A}\\mu(P),
\n\\]
\n
\nwhere $A$ is the fixed point set of $M$ under the lie group $S^1$ action, $\\mu$ is a function on $A$. \\section{Berline-Vergne Localization Formula}
\nLet $M$ be an even dimensional smooth closed oriented manifold. We assume that $M$ admits an $S^1$-(right)action. On $M$, $S^1$-(right)action is defined by a smooth map
\n\\[
\n\\phi:~M\\times S^1\\rightarrow M, \\quad \\phi(P,a)=Pa, \\text{for any }a\\in S^1
\n\\] such that for any $a,b\\in S^1$, $P\\in M$
\n\\begin{enumerate}
\n \\item $P1=\\phi(P,1)=P$;
\n \\item $(Pa)b=P(ab)$.
\n\\end{enumerate}
\nOne easily verifies the map $a:P\\in M\\rightarrow a(P)=Pa$ is a diffeomorphism for any $a\\in S^1$. We claim a riemannian metric $g^{TM}$ is \\emph{$S^1$-invariant}, if for any $a\\in S^1$ always have $a^*g^{TM}=g^{TM}$. Since $S^1$ is a compact Lie group, there always exists a $S^1$-invariant metric on $M$. In fact, if $g^{TM}$ is general riemannian metric, the following metric is $S^1$-invariant
\n\\[
\n\\tilde{g}^{TM}=\\int_{a\\in S^1}a^*g^{TM}.
\n\\]\\begin{excs}
\nPlease verify $\\tilde{g}^{TM}$ is $S^1$-invariant.
\n\\end{excs}Then we can assume that there exists a $S^1$-invariant riemannian metric $g^{TM}$ without loss of generally.<\/p>\n
For any $P\\in M$, the map $P:~S^1\\rightarrow M$, $P(a)=Pa$, gives a curve on $M$. By $S^1$ is a Lie group, using the exponent map this curve can be parameterize. Let $t\\in Lie(S^1)$ be a generator of the Lie algebra of $S^1$. then the above curve is
\n\\[
\n\\gamma_{_P}(\\epsilon)=\\phi(P,\\exp(\\epsilon t))=P\\exp(\\epsilon t).
\n\\]Take $K(P)=\\gamma’_{_P}(0)=(\\gamma_{_P})_*\\frac{\\partial}{\\partial \\epsilon}|_{\\epsilon=0}$. Then we obtain a vector field $K$ on $M$, and for any $f\\in C^{\\infty}(M)$,
\n\\[
\nK_{_P}f=\\left.\\frac{\\rd}{\\rd \\epsilon}f\\left(P\\exp(\\epsilon t)\\right)\\right|_{\\epsilon=0}, \\quad\\text{ for any }P\\in M.
\n\\]\\begin{excs}
\nProve the integral curve crossing $P$ of $K$ is $\\gamma_{_P}(\\epsilon)$.
\n\\end{excs}
\nBy the $S^1$-action preserves $g^{TM}$, $K$ is a \\emph{Killing vector field} on $M$(see[Chen, Li. Rie.Geo., Beijing university]). Let $\\nabla^E$ be Levi-Civita connection associated $g^{TM}$. Then for any $X,Y\\in \\Gamma(TM)$,
\n\\begin{equation}\\label{eq:2-1}
\n\\left\\langle \\nabla^{TM}_XK,Y \\right\\rangle + \\left\\langle \\nabla^{TM}_YK,X\\right\\rangle=0,
\n\\end{equation}i.e.,
\n\\[
\n\\left\\langle \\left(\\nabla^{TM}K\\right)(X),Y \\right\\rangle + \\left\\langle \\left(\\nabla^{TM}K\\right)(Y),X\\right\\rangle=0.
\n\\]It means $\\nabla^{TM}K$ is a antisymmetric operator on $\\Gamma(TM)$. Now we prove this fact. Let $\\mathcal{L}_K$ denote the Lie derivative of $K$ on $\\Gamma(TM)$. Since the $S^1$-action preserves $g^{TM}$ and the integral curve of $K$ is $\\gamma_{_P}$, $\\mathcal{L}_K$ also preserves $g^{TM}$, i.e., $\\mathcal{L}_Kg^{TM}=0$. For any $X,Y\\in \\Gamma(TM)$, one has
\n\\begin{align*}
\n\\left\\langle \\nabla^{TM}_XK,Y\\right\\rangle+\\left\\langle \\nabla^{TM}_YK,X\\right\\rangle
\n=&\\left\\langle [X,K]+ \\nabla^{TM}_KX,Y\\right\\rangle +\\left\\langle X , [Y,K]+\\nabla^{TM}_KY \\right\\rangle \\\\
\n=&\\left\\langle -\\mathcal{L}_KX,Y \\right\\rangle +\\left\\langle X,-\\mathcal{L}_KY \\right\\rangle +K\\left\\langle X,Y \\right\\rangle \\\\
\n=&\\left\\langle -\\mathcal{L}_KX,Y \\right\\rangle +\\left\\langle X,-\\mathcal{L}_KY \\right\\rangle +\\mathcal{L}_K\\left\\langle X,Y\\right\\rangle \\\\
\n=&\\left(\\mathcal{L}_Kg^{TM}\\right)(X,Y)=0.
\n\\end{align*}
\nIn fact, $K$ is a Killing vector field if and only if (1) holds.<\/p>\n
We still denote the Lei derivative of $K$ on $\\Omega^*(M)$ by $\\mathcal{L}_K$. The following \\emph{Cartan homotopy formula} on $\\Omega^*{M}$ is well-known,
\n\\[
\n\\mathcal{L}_K=\\rd i_{_K}+i_{_K}\\rd,
\n\\]where $i_{_K}:~\\Omega^*{M}\\rightarrow \\Omega^{*-1}{M}$,
\n\\[
\ni_{_K}(\\omega_1\\wedge\\cdots\\wedge\\omega_k)=\\sum_{j=1}^k(-1)^{j-1}\\omega_1\\wedge\\cdots\\widehat{\\omega}_j\\wedge\\cdots\\wedge\\omega_k
\n.\\]Let $\\Omega^*_K(M)=\\left\\{\\omega\\in \\Omega^*_K(M):\\mathcal{L}_K\\omega=0 \\right\\}$ be the subspace of \\emph{$\\mathcal{L}_K$-invariant form}. Set
\n\\[
\n\\rd _K=\\rd +i_{_k}: \\Omega^*(M)\\rightarrow \\Omega^*(M).
\n\\] Then, by $\\rd^2=0$ and $i_{_K}^2=0$ one has
\n\\[
\n\\rd^2_{K}=\\rd i_{_K}+i_{_K}\\rd=\\mathcal{L}_K.
\n\\] Thus $\\rd_K$ preserves $\\Omega^*_K(M)$ and $\\left.\\rd_K^2\\right|_{\\Omega^*_K(M)}=0$. The corresponding cohomology group
\n\\[
\nH^*_K(M)=\\frac{\\ker \\rd_{K}|_{\\Omega^*_K(M)}}{\\mathrm{Im} \\rd_{K}|_{\\Omega^*_K(M)}}
\n\\] is called the \\emph{$S^1$ equivariant cohomology} of $M$.<\/p>\n
Now, consider any element $\\omega\\in \\Omega^*(M)$. We say $\\omega$ is \\emph{$\\rd_K$-closed} if $\\rd_K\\omega=0$. The equivariant localization formula duo to Berline-Vergne(or Atiyah-Bott) shows that the integration of a $\\rd_K$-closed differential form over $M$ can be localized to the zero set of the Killing vector field $K$.
\n\\begin{excs}
\nProve that
\n\\[
\n\\mathrm{zero}\\{K\\}=\\emptyset\\Leftrightarrow\\text{ the fixed point set of }M\\text{ under }S^1\\text{-action is empty}. \\]\\end{excs}
\n\\begin{prop}\\label{pro:2-1}
\nIf $K$ has no zeros on $M$, then for any $\\omega\\in \\Omega^*(M)$ which is $\\rd_K$-closed, one has $\\int_M\\omega=0$.
\n\\end{prop}
\n\\begin{proof}
\nLet $\\theta\\in\\Omega^1(M)$ be the one form on $M$ such that for any $X\\in\\Gamma(TM)$, $i_{_K}\\theta=\\langle X,K\\rangle$. By $\\mathcal{L}_K$ preserves $g^{TM}$, one has
\n\\[
\n \\mathcal{L}_K\\big(\\theta(X)\\big)=\\mathcal{L}_K\\big(i_{_X}\\theta\\big)=\\mathcal{L}_K\\langle X,K\\rangle.
\n\\]i.e.,
\n\\begin{align*}
\n\\left(\\mathcal{L}_K\\theta\\right)(X)+\\theta\\left(\\mathcal{L}_KX\\right)=&\\langle \\mathcal{L}_KK,X \\rangle+\\langle \\mathcal{L}_KX,K \\rangle\\\\
\n=&i_{\\mathcal{L}_KX}\\theta=\\theta\\left(\\mathcal{L}_KX\\right).
\n\\end{align*}
\nThus $\\mathcal{L}_K\\theta=0$. one then sees that $\\rd_K\\theta$ is $\\rd_K$-closed.
\nBy $\\omega$ is $\\rd_K$-closed, for any $T\\geq0$ one has
\n\\begin{align*}
\n\\int_M\\omega\\exp(-T\\rd_K\\theta)=&\\int_M\\omega\\left[1+\\sum_{i=1}^{\\infty}\\frac{(-1)^i}{i!}T^i(\\rd_K\\theta)^i\\right]\\\\
\n=&\\int_M\\omega + \\int_M\\omega\\wedge\\rd_K\\left[\\sum_{i=1}^{\\infty}\\frac{(-1)^i}{i!}T^i\\theta\\wedge(\\rd_K\\theta)^{i-1}\\right]\\\\
\n=&\\int_M\\omega +(-1)^{\\mathrm{deg}(\\omega)} \\int_M\\rd_K\\left[\\omega\\wedge\\sum_{i=1}^{\\infty}\\frac{(-1)^i}{i!}T^i\\theta\\wedge(\\rd_K\\theta)^{i-1}\\right]\\\\
\n=&\\int_M\\omega.
\n\\end{align*}
\nOtherwise,
\n\\begin{align*}
\n\\int_M\\omega\\exp(-T\\rd_k\\theta)=&\\int_M\\omega\\exp\\left[-T(\\rd\\theta+i_K\\theta)\\right]\\\\
\n=&\\int_M\\omega\\exp\\left[-T(\\rd\\theta+|k|^2)\\right]\\\\
\n=&\\int_M\\Big(\\omega\\exp(-T|K|^2)\\Big)\\left[\\sum_{i=1}^{\\dim M\/2}\\frac{(-1)^i}{i!}T^i(\\rd\\theta)^i \\right].
\n\\end{align*}
\nAnd, as $K$ has no zeros on $M$, $|K|$ has a positive lower bound $\\delta>0$ on $M$. By $M$ is closed, one sees easily that
\n\\[
\n\\int_M\\Big(\\omega\\exp(-T|K|^2)\\Big)\\left[\\sum_{i=1}^{\\dim M\/2}\\frac{(-1)^i}{i!}T^i(\\rd\\theta)^i\\right]\\rightarrow 0, \\quad\\text{ as }T\\rightarrow0.
\n\\]Thus
\n\\[
\n\\int_M\\omega=0.
\n\\]\\end{proof}
\nIn the previous discussion we considered the case of $\\mathrm{zero}(K)=\\emptyset$. Now we assume the zero set of $K$ is discrete.<\/p>\n
For every $p\\in \\mathrm{zero}(K)$, there is a small open neighborhood $U_p$ of $p$ and an oriented coordinate system $(x^1,\\cdots,x^{2l})$ with $2l=\\dim M$ such that we have
\n\\[
\n\\left.g^{TM}\\right|_{U_p}=\\left(\\rd x^1\\right)^2+\\cdots+\\left(\\rd x^{2l}\\right)^2
\n\\]and
\n\\[
\nK|_{U_p}=\\sum_{i=1}^{l}\\lambda_i\\left(x^{2i}\\frac{\\partial}{\\partial x^{2i-1}}-x^{2i-1}\\frac{\\partial}{\\partial x^{2i}}\\right)
\n\\]with each $\\lambda_i\\neq0$ for $1\\leq i\\leq l$.<\/p>\n
Set
\n\\[
\n\\lambda(p)=\\lambda_1\\cdots\\lambda_l.
\n\\]
\nIn fact, the existence of $U_p$ is not trivial. But, I am very sorry about that I don’t understand about the existence.<\/p>\n
\\begin{thm}
\nIf $\\mathrm{zero}(K)$ is discrete, then for any $\\rd _K$-closed form $\\omega\\in\\Omega^*(M)$, one has
\n\\[
\n\\int_M\\omega=(2\\pi)^l\\sum_{p\\in\\mathrm{zero}(K)}\\frac{\\omega^{[0]}(p)}{\\lambda(p)}.
\n\\]
\n\\end{thm}
\n\\begin{proof}
\nSince $M\\setminus \\cup_{p\\in\\mathrm{zero}(K)}U_p$ is closed manifold and $K$ has no zeros on $M\\setminus \\cup_{p\\in\\mathrm{zero}(K)}U_p$, using the proposition 1 we have
\n\\[
\n\\int_{M\\setminus \\cup_{p\\in\\mathrm{zero}(K)}U_p}\\omega=\\int_{M\\setminus \\cup_{p\\in\\mathrm{zero}(K)}U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right)=0.
\n\\]Hence
\n\\begin{align*}
\n\\int_M\\omega=&\\int_{M\\setminus \\cup_{p\\in\\mathrm{zero}(K)}U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right)
\n+\\sum_{p\\in\\mathrm{zero}(K)}\\int_{U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right)\\\\
\n=&\\sum_{p\\in\\mathrm{zero}(K)}\\int_{U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right).
\n\\end{align*}
\nOn $U_p$ we have
\n\\[
\n|K|^2=\\sum_{i=1}^{l}\\lambda_i^2\\left[(x^{2i})^2+(x^{2i-1})^2\\right],\\quad
\n\\theta=\\sum_{i=1}^{l}\\lambda_i\\left(x^{2i}\\rd x^{2i-1}-x^{2i-1}\\rd x^{2i}\\right).
\n\\]Then,
\n\\begin{align*}
\n&\\int_{U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right)\\\\
\n&=\\int_{U_p}\\omega\\exp\\left(-T|K|^2-T\\rd\\theta\\right)\\\\
\n&=\\int_{U_p}\\omega\\exp\\left[-T\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\n\\exp\\left(2T\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)\\\\
\n&=\\int_{U_p}\\exp\\left[-T\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\n\\omega\\sum_{k=0}^{\\infty}\\left(2T\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^k\\\\
\n&=\\int_{U_p}\\exp\\left[-T\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\n\\sum_{j=0}^{l}\\omega^{[2j]}\\left(2T\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^{l-j}\\\\
\n\\end{align*}
\nMake the change of the coordinate system $(x^1,\\cdots,x^{2l})\\rightarrow\\sqrt{T}(x^1,\\cdots,x^{2l})$. Above integral is rewritten
\n\\begin{align*}
\n\\int_{\\sqrt{T}U_p}&\\exp\\left[-T\\sum_{i=1}^{l}\\lambda_i^2\\frac{1}{T}\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]\\\\
\n&\\sum_{j=0}^{l}T^{-j}\\omega^{[2j]}\\left(\\frac{x}{\\sqrt{T}}\\right)2^{l-j}\\left(T\\sum_{i=1}^l\\lambda_iT^{-1}\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^{l-j}\\\\
\n=\\int_{\\sqrt{T}U_p}&\\exp\\left[-\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]\\\\
\n&\\sum_{j=0}^{l}T^{-j}\\omega^{[2j]}\\left(\\frac{x}{\\sqrt{T}}\\right)2^{l-j}\\left(\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^{l-j}
\n\\end{align*}
\nWhen $0<j\\leq l$, using $\\int_{R}\\exp{(-x^2)}\\rd x=\\sqrt{\\pi}$ one can easily find that
\n\\begin{align*}
\n\\int_{\\sqrt{T}U_p}&\\exp\\left[-\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\nT^{-j}\\omega^{[2j]}\\left(\\frac{x}{\\sqrt{T}}\\right)2^{l-j}\\left(\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^{l-j}\\\\
\n&\\rightarrow 0 \\quad as\\quad T\\rightarrow+\\infty.
\n\\end{align*}
\nThus,
\n\\begin{align*}
\n&\\int_{U_p}\\omega\\exp\\left(-T\\rd_K\\theta\\right)\\\\
\n&=\\int_{\\sqrt{T}U_p}\\exp\\left[-\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\n\\omega^{[0]}\\left(\\frac{x}{\\sqrt{T}}\\right)2^{l}\\left(\\sum_{i=1}^l\\lambda_i\\rd x^{2i-1}\\wedge\\rd x^{2i}\\right)^{l}\\\\
\n&=\\int_{\\sqrt{T}U_p}\\exp\\left[-\\sum_{i=1}^{l}\\lambda_i^2\\left((x^{2i-1})^2+(x^{2i})^2\\right)\\right]
\n\\omega^{[0]}\\left(\\frac{x}{\\sqrt{T}}\\right)2^{l}
\n\\lambda(p)^{-1}\\rd\\lambda_1x^1\\rd\\lambda_1x^2\\cdots\\rd\\lambda_lx^{2l}\\\\
\n&\\rightarrow(2\\pi)^l\\lambda(p)^{-1}\\omega^{[0]}(0)\\quad as\\quad T\\rightarrow+\\infty.
\n\\end{align*}
\nthis completes the proof.
\n\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"
In Chapter One we have defined character …<\/p>\n