{"id":2118,"date":"2012-03-08T23:50:45","date_gmt":"2012-03-08T15:50:45","guid":{"rendered":"http:\/\/wamath.sinaapp.com\/?p=2118"},"modified":"2012-03-08T23:50:45","modified_gmt":"2012-03-08T15:50:45","slug":"adiabatic-limit-and-the-bott-connection","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2118","title":{"rendered":"Adiabatic Limit and the Bott Connection"},"content":{"rendered":"

Background<\/strong>
\nLet $M$ be an even dimensional oriented closed spin manifold. We define the \\emph{$\\hat{A}$-genus} of $M$, denote by $\\hat{A}(M)$, by
\n\\[
\n\\hat{A}(M)=\\big\\langle \\hat{A}(TM),[M] \\big\\rangle= \\int_M\\hat{A}(TM,\\nabla^{TM})\\in\\Z.
\n\\]
\nIf there exists a Riemannian metric $g^{TM}$ such that the related section curvature $K>0$, then $\\hat{A}(M)=0$.
\n\\begin{proof}
\nSince
\n\\[
\nD^2=-\\triangle+\\frac{K}{4}
\n\\]
\nis a non-negative defined operator and $-\\triangle$, $\\frac{K}{4}$ too. Then if $D^2s=0$ for any $s\\in C^{\\infty}(M)$, \\[
\n0=\\langle\\langle D^2s,s\\rangle\\rangle=\\langle\\langle Ds,Ds \\rangle\\rangle=\\langle\\langle-\\triangle s,s \\rangle\\rangle+\\langle\\langle \\frac{K}{4}s,s \\rangle\\rangle \\geq0.
\n\\]
\nThus $s=0$.
\n\\end{proof}<\/p>\n

In fact, if we change the condition $K>0$ into $K\\geq0$, the result is still right. Because there is an other Riemannian metric $\\widetilde{g}^{TM}$ such that the related section curvature $\\widetilde{K}>0$ by the theory of Yamabe problem.
\nWe consider the above question under a weak condition. If the integral manifold of any integrable sub-bundle of $TM$ is a spin manifold, could I still obtain $\\hat{A}(M)=0$? Recently, Weiping Zang settles this question when the integral manifold of any integrable sub-bundle of $TM$ is almost riemannian. This section comes from his related research.<\/p>\n

Adiabatic Limit<\/strong>
\nOn $F^{\\bot}$, there are two connection $\\nabla^{F^{\\bot}}$, $\\widetilde{\\nabla}^{F^{\\bot}}$. Obviously, the connection $\\nabla^{F^{\\bot}}$ is preserved metric, connection $\\widetilde{\\nabla}^{F^{\\bot}}$ is not preserved metric by definition1.13(i). In fact, by passing $g^{TM}$ to its adiabatic limit, one sees that underlying limit of $\\nabla^{F^{\\bot}}$ and the Bott connection $\\widetilde{\\nabla}^{F^{\\bot}}$ are ultimately related.<\/p>\n

For any $\\epsilon>0$, let $g^{TM,\\epsilon}$ be the metric on $TM$ defined by
\n\\[
\n g^{TM,\\epsilon}=g^{F}\\oplus \\frac{1}{\\epsilon}g^{F^{\\bot}}.
\n\\]
\nLet $\\nabla^{TM,\\epsilon}$ be the Levi-Civita connection of $g^{TM,\\epsilon}$. Let $\\nabla^{F,\\epsilon}$ (resp. $\\nabla^{F^{\\bot},\\epsilon}$) be the restriction of $\\nabla^{TM,\\epsilon}$ to $F$ (resp. $F^{\\bot}$). The process of taking the limit $\\epsilon\\rightarrow0$ is called taking the \\emph{adiabatic limit}.<\/p>\n

In fact, as $\\epsilon\\rightarrow0 $ the distance between leafs of foliation foliated by $F$ in direction of $F^{\\bot}$ increases gradually. We will examine the behavior of $\\nabla^{f^{\\bot},\\epsilon}$ as $\\epsilon$. Let $\\widetilde{\\nabla}^{f^{\\bot},*}$ be the connection on $F{\\bot}$ which is dual to $\\widetilde{\\nabla}^{F^{\\bot}}$. That is , for any sections $U,V$$\\in\\Gamma(F^{\\bot})$,
\n\\[
\nd\\langle U,V \\rangle_{g^{TM}}=\\left\\langle \\widetilde{\\nabla}^{F^{\\bot}}U,V\\right\\rangle_{g^{TM}} + \\left\\langle \\widetilde{\\nabla}^{F^{\\bot},*}U,V\\right\\rangle_{g^{TM}}.
\n\\]
\n\\begin{excs}
\nValidate $\\widetilde{\\nabla}^{F^{\\bot},*}$ is a connection on $F^{\\bot}$.
\n\\end{excs}
\nSet
\n\\[
\n \\omega^{F^{\\bot}}= \\widetilde{\\nabla}^{F^{\\bot},*}-\\widetilde{\\nabla}^{F^{\\bot}};\\quad
\n \\hat{\\nabla}^{F^{\\bot}}=\\widetilde{\\nabla}^{F^{\\bot}}+\\frac{\\omega^{F^{\\bot}}}{2}.
\n\\]
\nOne verifies easily that the connection $\\hat{\\nabla}^{F^{\\bot}}$ preserves $g^{F^{\\bot}}$ by the definition of dual connection, and $\\widetilde{\\nabla}^{F^{\\bot}}$ preserves $g^{F^{\\bot}}$ when for any $X\\in\\Gamma(F)$, $\\omega^{F^{\\bot}}(X)=0$.
\n\\begin{thm}
\nFor any smooth section $X\\in\\Gamma(F)$. one has
\n\\[
\n \\lim_{\\epsilon\\rightarrow0}\\nabla^{F^{\\bot},\\epsilon}_{X}=\\hat{\\nabla}^{F^{\\bot}}_{X}.
\n\\]
\n\\end{thm}
\n\\begin{proof}
\nWe only need to prove that for any $U,V\\in\\Gamma(F^{\\bot})$,
\n\\[
\n \\left\\langle \\nabla^{F^{\\bot},\\epsilon}_{X}U,V\\right\\rangle_{g^{TM}}\\rightarrow \\left\\langle\\hat{\\nabla}^{F^{\\bot}}_{X}U,V\\right\\rangle_{g^{TM}},\\quad as\\quad \\epsilon\\rightarrow0.
\n\\]
\nBy the definition of Bott connection and $\\hat{\\nabla}^{F^{\\bot}}=\\frac{1}{2}\\left(\\widetilde{\\nabla}^{F^{\\bot},*}+\\widetilde{\\nabla}^{F^{\\bot}}\\right)$, we have
\n\\begin{align*}
\n&\\left\\langle \\nabla^{F^{\\bot},\\epsilon}_{X}U,V \\right\\rangle_{g^{TM,\\epsilon}}\\\\
\n=&\\frac{1}{2}\\Big\\{ X\\langle U,V \\rangle_{g^{TM,\\epsilon}} +U\\langle X,V\\rangle_{g^{TM,\\epsilon}} -V\\langle X,U\\rangle_{g^{TM,\\epsilon}} \\\\
\n &+\\langle [X,U],V\\rangle_{g^{TM,\\epsilon}}-\\langle[U,V],X\\rangle_{g^{TM,\\epsilon}}+\\langle[V,X],U\\rangle_{g^{TM,\\epsilon}}\\Big\\}\\\\
\n=&\\frac{1}{2\\epsilon}\\Big\\{ X\\langle U,V\\rangle_{g^{TM}} + \\langle [X,U],V\\rangle_{g^{TM}} – \\langle[X,V],U\\rangle_{g^{TM}}\\Big\\}-\\frac{1}{2}\\langle[U,V],X\\rangle_{g^{TM}}\\\\
\n=&\\frac{1}{2\\epsilon}\\Big\\{ X\\langle U,V\\rangle_{g^{TM}} + \\langle p^{\\bot}[X,U],V\\rangle_{g^{TM}} – \\langle p^{\\bot}[X,V],U\\rangle_{g^{TM}} \\Big\\}-\\frac{1}{2}\\langle[U,V],X\\rangle_{g^{TM}}\\\\
\n=&\\frac{1}{2\\epsilon}\\Big\\{ X\\langle U,V\\rangle_{g^{TM}} + \\langle \\widetilde{\\nabla}^{F^{\\bot}}_XU,V\\rangle_{g^{TM}} – \\langle \\widetilde{\\nabla}^{F^{\\bot}}_XV,U\\rangle_{g^{TM}} \\Big\\}-\\frac{1}{2}\\langle[U,V],X\\rangle_{g^{TM}}\\\\
\n=&\\frac{1}{2\\epsilon}\\Big\\{ \\langle \\widetilde{\\nabla}^{F^{\\bot}}_XU,V\\rangle_{g^{TM}} + \\langle \\widetilde{\\nabla}^{F^{\\bot},*}_XV,U\\rangle_{g^{TM}} \\Big\\}-\\frac{1}{2}\\langle[U,V],X\\rangle_{g^{TM}}\\\\
\n=&\\frac{1}{\\epsilon} \\left\\langle \\hat{\\nabla}^{F^{\\bot}}_XU,V\\right\\rangle_{g^{TM}} -\\frac{1}{2}\\langle[U,V],X\\rangle_{g^{TM}},
\n\\end{align*}
\nand
\n\\[
\n\\left\\langle \\nabla^{F^{\\bot},\\epsilon}_{X}U,V \\right\\rangle_{g^{TM,\\epsilon}}=\\frac{1}{\\epsilon}\\left\\langle \\nabla^{F^{\\bot},\\epsilon}_{X}U,V \\right\\rangle_{g^{TM}}.
\n\\]
\nHence,
\n\\[
\n\\left\\langle \\nabla^{F^{\\bot},\\epsilon}_{X}U,V \\right\\rangle_{g^{TM}}=\\left\\langle \\hat{\\nabla}^{F^{\\bot}}_XU,V\\right\\rangle_{g^{TM}} -\\frac{1}{2}\\epsilon\\langle[U,V],X\\rangle_{g^{TM}}.
\n\\]
\nThis ends the proof.
\n\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"

Background Let $M$ be an even dimensiona …<\/p>\n

Adiabatic Limit and the Bott Connection<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,12],"tags":[],"class_list":["post-2118","post","type-post","status-publish","format-standard","hentry","category-index-theory","category-mathnotes"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2118"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2118\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}