{"id":740,"date":"2020-02-11T10:00:10","date_gmt":"2020-02-11T10:00:10","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=740"},"modified":"2020-02-11T10:00:10","modified_gmt":"2020-02-11T10:00:10","slug":"zixuanjihejianjie","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=740","title":{"rendered":"\u81ea\u65cb\u51e0\u4f55\u7b80\u4ecb"},"content":{"rendered":"

\u81ea\u65cb\u51e0\u4f55\u7b80\u4ecb<\/span>\nVan Abel<\/span>\n02\/11\/2020<\/span><\/p>\n


\n\n

\u6458\u8981<\/span>.<\/span> \u6211\u4eec\u7b80\u5355\u7684\u4ecb\u7ecd\u81ea\u65cb\u51e0\u4f55\u91cc\u57fa\u672c\u7684\u6982\u5ff5\uff0c\u53ef\u4ee5\u53c2\u8003[1<\/a>]\u3002
\n<\/div>
\n
\u5b9a\u4e49 1<\/span> (\u81ea\u65cb\u7ed3\u6784<\/span>).<\/span> \u5047\u8bbe$(M^m,h)$\u662f\u4e00\u4e2a\u53ef\u5b9a\u5411\u9ece\u66fc\u6d41\u5f62\u3002 $M$\u4e0a\u7684\u4e00\u4e2a\u81ea\u65cb\u7ed3\u6784<\/span>\u662f\u4e00\u4e2a\u4e8c\u5143\u5bf9$(\\mathrm{Spin}(M),\\eta)$\uff0c\u5176\u4e2d$\\mathrm{Spin}(M)$\u662f$M$\u4e0a\u4e00\u4e2a$\\mathrm{Spin}_m$-\u4e3b\u4e1b\uff0c\u800c$\\eta$\u662f\u4e00\u4e2a2\u91cd\u8986\u76d6\u4f7f\u5f97\u4e0b\u56fe\u4ea4\u6362
\n\\[
\n\\begin{CD}
\n\\mathrm{Spin}(M)\\times \\mathrm{Spin}_m@>\\rho_1>> \\mathrm{Spin}(M)@>\\pi>> M\\\\
\n@VV \\eta\\times \\mathrm{Ad}V@VV\\eta V@|\\\\
\n\\mathrm{SO}(M)\\times \\mathrm{SO}_m@>\\rho_2>> \\mathrm{SO}(M)@>>\\pi>M,
\n\\end{CD}
\n\\]
\n\u5176\u4e2d$\\rho_1$, $\\rho_2$\u5206\u522b\u662f\u81ea\u65cb\u7fa4$\\mathrm{Spin}_m$\u548c$\\mathrm{SO}_m$\u4f5c\u7528\u5728\u4e3b\u4e1b$\\mathrm{Spin}(M)$\u548c$\\mathrm{SO}(M)$\u4e0a\u3002
\n<\/div>
\n$M$\u4e0a\u81ea\u65cb\u7ed3\u6784\u7684\u5b58\u5728\u6027\u7b49\u4ef7\u4e8e$M$\u7684\u7b2c\u4e8cStiefel-Whitney\u7c7b$\\omega_2(M)=0$\uff0c\u8fd9\u662f\u4e00\u4e2a\u62d3\u6251\u9650\u5236\u3002
\n
\n
\u5b9a\u4e49 2<\/span> (\u65cb\u91cf\u4e1b<\/span>).<\/span> \u5047\u8bbe$M$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62\uff0c\u5173\u4e8e\u5176\u4e0a\u81ea\u65cb\u7ed3\u6784$\\mathrm{Spin}(M)$\u7684\uff08\u590d\uff09\u65cb\u91cf\u4e1b<\/span>\u5b9a\u4e49\u4e3a
\n\\[
\n\\Sigma M \\mathpunct{:}=\\mathrm{Spin}(M)\\times_\\rho\\Sigma_m,
\n\\]
\n\u5176\u4e2d$\\rho \\mathpunct{:} \\mathrm{Spin}_n\\to \\mathrm{Aut}(\\Sigma_m)$\u662f\u81ea\u65cb\u7fa4$\\mathrm{Spin}_m$\u7684\uff08\u590d\uff09\u8868\u793a\u3002
\n<\/div>
\n\u5728\u65cb\u91cf\u4e1b$\\Sigma M$\u4e0a\uff0c\u6211\u4eec\u6709Clifford\u5173\u7cfb
\n\\[
\nX\\cdot Y\\cdot \\psi+Y\\cdot X\\cdot \\psi=-2h(X,Y)\\psi,\\quad\\forall X,Y\\in\\Gamma(TM),\\,\\psi\\in\\Gamma(\\Sigma M),
\n\\]
\n\u8fd9\u91cc$h$\u662f$M$\u4e0a\u7684\u5ea6\u91cf\u3002\u5982\u679c\u6211\u4eec\u9009\u62e9$\\Sigma M$\u4e0a\u7684\u8bf6\u7c73\u5c14\u7279\u5ea6\u91cf\uff0c\u8fd9Clifford\u4e58\u6cd5\u662f\u53cd\u5bf9\u79f0\u7684\uff0c\u5373
\n\\[
\n\\left\\langle X\\cdot\\psi_1,\\psi_2 \\right\\rangle+\\left\\langle \\psi_1,X\\cdot\\psi_2 \\right\\rangle=0,\\quad\\forall X\\in\\Gamma(TM),\\,\\psi_1,\\psi_2\\in\\Gamma(\\Sigma M).
\n\\]
\n\u5229\u7528$M$\u4e0a\u7684\u9ece\u66fc\u8054\u7edc\uff0c\u53ef\u4ee5\u8bf1\u5bfc$\\Sigma M$\u4e0a\u7684\u4e00\u4e2a\u8054\u7edc$\\nabla^{\\Sigma M}$, \u5982\u679c\u5c06\u5176\u5bf9\u5e94\u7684\u66f2\u7387\u8bb0\u4e3a$R^{\\Sigma M}$, \u5219\u6709\u5982\u4e0b\u57fa\u672c\u5173\u7cfb.
\n
\u547d\u9898 3<\/span> (\u65cb\u91cf\u4e1b\u4e0a\u8054\u7edc\u4e0e\u66f2\u7387\u7684\u5c40\u90e8\u8868\u793a<\/span>).<\/span> \u65cb\u91cf\u4e1b$\\Sigma M$\u4e0a\u7684\u5171\u53d8\u5fae\u5206\u5c40\u90e8\u53ef\u8868\u793a\u4e3a
\n\\[
\n\\nabla^{\\Sigma M}\\psi=\\frac{1}{4}h(\\nabla e_{\\alpha},e_{\\beta})e_\\alpha\\cdot e_\\beta\\cdot \\psi;
\n\\]
\n\u5982\u679c\u6211\u4eec\u5c06$M$\u4e0a\u7684\u9ece\u66fc\u66f2\u7387\u8bb0\u4e3a$R^M$, \u5219
\n\\[
\nR^{\\Sigma M}(X,Y)\\psi=\\frac{1}{4}h\\left( R^{M}(X,Y)e_\\alpha,e_\\beta \\right)e_\\alpha\\cdot e_\\beta\\cdot\\psi.
\n\\]
\n\u8fd9\u91cc\uff0c$X,Y\\in\\Gamma(TM)$, $\\left\\{ e_\\alpha \\right\\}$\u662f$TM$\u7684\u4e00\u4e2a\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6\u3002
\n<\/div>
\n\u4e8b\u5b9e\u4e0a\uff0c\u65cb\u91cf\u4e1b\u4e0a\u7684\u5171\u53d8\u5fae\u5206\u8fd8\u548c\u4e1b\u5ea6\u91cf\u4ee5\u53caClifford\u4e58\u6cd5\u90fd\u76f8\u5bb9\uff1a
\n
\u547d\u9898 4<\/span>.<\/span> \u5047\u8bbe\u8bb0\u53f7\u5982\u4e0a\uff0c\u5219
\n\\begin{align*}
\n\\partial_X\\left\\langle \\psi_1,\\psi_2 \\right\\rangle_{\\Sigma M}&=\\left\\langle \\nabla_{X}^{\\Sigma M}\\psi_1,\\psi_2 \\right\\rangle_{\\Sigma M}+\\left\\langle \\psi_1,\\nabla_X^{\\Sigma M}\\psi_2 \\right\\rangle_{\\Sigma M},\\\\
\n\\nabla_{X}^{\\Sigma M}(Y\\cdot\\psi)&=(\\nabla_XY)\\cdot\\psi+Y\\cdot\\nabla_{X}^{\\Sigma M}\\psi.
\n\\end{align*}
\n<\/div>
\n\u6211\u4eec\u5c06\u72c4\u62c9\u514b\u7b97\u5b50\u5b9a\u4e49\u5982\u4e0b\uff1a
\n
\u5b9a\u4e49 5<\/span>.<\/span> \u72c4\u62c9\u514b\u7b97\u5b50\u5b9a\u4e49\u4e3a\u4f5c\u7528\u5728\u65cb\u91cf\u4e1b$\\Sigma M$\u622a\u9762\u4e0a\u7684\u5171\u53d8\u5fae\u5206Clifford\u4e58\u6cd5\u7684\u590d\u5408\uff0c\u5c40\u90e8\u5730
\n\\[
\n\\not\\partial\\psi \\mathpunct{:}=e_\\alpha\\cdot\\nabla_{e_\\alpha}^{\\Sigma M}\\psi.
\n\\]
\n<\/div>
\n
\u5f15\u7406 6<\/span> (\u72c4\u62c9\u514b\u7b97\u5b50\u7684\u57fa\u672c\u6027\u8d28<\/span>).<\/span> \u72c4\u62c9\u514b\u7b97\u5b50\u662f\u4e00\u9636\u692d\u5706\u504f\u5fae\u5206\u7b97\u5b50\uff0c\u5b83\u6ee1\u8db3\u4ee5\u4e0b\u57fa\u672c\u6027\u8d28\uff1a
\n
  1. \u5b83\u662f\u5f31\u692d\u5706\u7684\uff1b<\/li>
  2. \u5bf9\u7d27\u6d41\u5f62$M$\u4e0a\u7684$L^2$\u5185\u79ef\uff0c\u5b83\u662f\u81ea\u5bf9\u5076\u7684\uff1b<\/li>
  3. \u5982\u4e0b\u7684Schr\\”odinger–Lichnerowicz\u516c\u5f0f\u6210\u7acb
    \n\\[
    \n\\not\\partial^2=\\nabla^*\\nabla+\\frac{1}{4}R,
    \n\\]
    \n\u5176\u4e2d$\\nabla$\u662f$M$\u4e0a\u7684\u9ece\u66fc\u8054\u7edc\uff0c$R$\u662f$M$\u7684\u6570\u91cf\u66f2\u7387\u3002<\/li><\/ol><\/div>
    \n\n<\/p>\n
    \n\t
    \u53c2\u8003\u6587\u732e<\/div>\n\t
    1. O. Hijazi<\/span>, Spectral properties of the Dirac operator and geometrical structures<\/a>, World Sci. Publ., River Edge, NJ<\/span>, 2001<\/span>. 116---169<\/span>. MR1867733<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"

      \u81ea\u65cb\u51e0\u4f55\u7b80\u4ecb Van Abel 02\/11\/2020 \\begin{document} \u6458\u8981. \u6211\u4eec\u7b80\u5355\u7684\u4ecb\u7ecd… \u7ee7\u7eed\u9605\u8bfb\u81ea\u65cb\u51e0\u4f55\u7b80\u4ecb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-740","post","type-post","status-publish","format-standard","hentry","category-math","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/740","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=740"}],"version-history":[{"count":4,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/740\/revisions"}],"predecessor-version":[{"id":744,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/740\/revisions\/744"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=740"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=740"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=740"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}