{"id":467,"date":"2017-12-07T14:48:51","date_gmt":"2017-12-07T14:48:51","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=467"},"modified":"2017-12-12T02:57:29","modified_gmt":"2017-12-12T02:57:29","slug":"xiancongshangdelianluoqushuaihelequnyijichenlei","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=467","title":{"rendered":"\u7ebf\u4e1b\u4e0a\u7684\u8054\u7edc\u3001\u66f2\u7387\u3001\u548c\u4e50\u7fa4\u4ee5\u53ca\u9648\u7c7b"},"content":{"rendered":"<p><span id=\"contents\"  style=\"text-align:center; font-size:18px; font-variant:small-caps;display:block;\">\u76ee\u5f55<\/span><br \/>\n          <span id=\"sec:content\"><a href=\"#contents\">\u76ee\u5f55<\/a><\/span><br \/><span>&#x00A0;1.&#x00A0;&#x00A0;<a href=\"#sec:1\">\u7ebf\u4e1b\u7684\u5b9a\u4e49<\/a><\/span><br \/><span>&#x00A0;2.&#x00A0;&#x00A0;<a href=\"#sec:2\">\u7ebf\u4e1b\u7684\u8054\u7edc<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.1.&#x00A0;&#x00A0;<a href=\"#sec:2.1\">\u8054\u7edc\u7684\u5c40\u90e8\u8868\u793a<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.2.&#x00A0;&#x00A0;<a href=\"#sec:2.2\">\u8054\u7edc\u7684\u8f6c\u79fb\u5173\u7cfb<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;2.3.&#x00A0;&#x00A0;<a href=\"#sec:2.3\">\u8054\u7edc\u8bf1\u5bfc\u7684\u5171\u53d8\u5916\u5fae\u5206<\/a><\/span><br \/><span>&#x00A0;3.&#x00A0;&#x00A0;<a href=\"#sec:3\">\u7ebf\u4e1b\u7684\u66f2\u7387<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;3.1.&#x00A0;&#x00A0;<a href=\"#sec:3.1\">\u66f2\u7387\u7684\u57fa\u672c\u6027\u8d28<\/a><\/span><br \/><span>&#x00A0;4.&#x00A0;&#x00A0;<a href=\"#sec:4\">Hermitian\u7ebf\u4e1b<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;4.1.&#x00A0;&#x00A0;<a href=\"#sec:4.1\">Hermitian\u5168\u7eaf\u7ebf\u4e1b\u7684\u66f2\u7387<\/a><\/span><br \/><span>&#x00A0;5.&#x00A0;&#x00A0;<a href=\"#sec:5\">\u7ebf\u4e1b\u7684\u548c\u4e50\u7fa4<\/a><\/span><br \/><span>&#x00A0;&#x00A0;&#x00A0;5.1.&#x00A0;&#x00A0;<a href=\"#sec:5.1\">\u548c\u4e50\u7fa4\u4e0e\u66f2\u7387\u7684\u5173\u7cfb<\/a><\/span><br \/><span>&#x00A0;6.&#x00A0;&#x00A0;<a href=\"#sec:6\">\u9648\u7c7b<\/a><\/span><br \/><br \/>\n<span class=\"latex_section\">1.&#x00A0;\u7ebf\u4e1b\u7684\u5b9a\u4e49<a id=\"sec:1\"><\/a><\/span>\n\n\u7ebf\u4e1b\u662f\u5411\u91cf\u4e1b\u7684\u6700\u7b80\u5355\u7684\u5b9e\u4f8b\u3002<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 1<\/span><span class='latex_defn_h'>.<\/span> \u6d41\u5f62$M$(\u4e0d\u4e00\u5b9a\u590d)\u4e0a\u7684\u4e00\u4e2a(\u590d)\u7ebf\u4ece$(L,\\pi)$, \u8fd9\u91cc$L$\u662f\u4e00\u4e2a\u6d41\u5f62, $\\pi:L\\to M$\u662f\u5149\u6ed1\u6ee1\u5c04, \u4f7f\u5f97<br \/>\n<ul><li>\u6bcf\u4e2a\u7ea4\u7ef4$L_m:=\\pi^{-1}(m)$\u662f\u4e00\u4e2a1\u7ef4(\u590d)\u7ebf\u6027\u7a7a\u95f4\uff1b<\/li><li>\u5c40\u90e8\u5e73\u51e1\uff1a\u5bf9\u4efb\u610f\u7684$m\\in M$, \u5b58\u5728$M$\u7684\u5f00\u90bb\u57df$U\\ni m$\u4ee5\u53ca\u5149\u6ed1\u5fae\u5206\u540c\u80da$\\phi:\\pi^{-1}(U)\\to U\\times\\mathbb{C}$, \u4f7f\u5f97$\\phi(L_m)=\\{m\\}\\times \\mathbb{C}$\u4e14$\\phi|_{L_m}$\u662f\u4e00\u4e2a\u7ebf\u6027\u540c\u6784\u3002<\/li><\/ul><\/div><!--more--><\/p>\n<p><div class='latex_rmk'><span class='latex_rmk_h'>\u6ce8\u8bb0 1<\/span><span class='latex_rmk_h'>.<\/span> \u6309\u7167\u5b9a\u4e49\u53ef\u4ee5\u9a8c\u8bc1, $\\pi$\u662f\u4e00\u4e2a\u6df9\u6ca1(submersion), \u5373$\\pi_*|_p\uff1aTL\\to T_{\\pi(p)}M$\u662f\u6ee1\u5c04\u3002<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 1<\/span> (<span class='latex_examp_name'>Hopf\u4e1b<\/span>)<span class='latex_examp_h'>.<\/span> \u56de\u5fc6, $\\mathbb{C}P^1$\u7684\u9f50\u6b21\u5750\u6807\u4e3a$z=[z^0,z^1]$, \u5b83\u53ef\u7531\u4e24\u4e2a\u5750\u6807\u5361\u8986\u76d6$\\{(U_i,\\psi_i)\\}_{i=0}^1$, \u5176\u4e2d$U_1=\\{[z]:z^i\\neq0\\}$, $\\psi_0([z])=z^1\/z_0$, $\\psi_1([z])=z^0\/z^1$. $\\mathrm{C}P^1$\u548c$S^2$\u662f\u5fae\u5206\u540c\u80da\u7684, \u540c\u80da\u6620\u5c04\u53ef\u7531<br \/>\n\\begin{align*}<br \/>\n\\Psi:S^2&#038;\\to \\mathbb{C}P^1\\\\<br \/>\n(x,y,z)&#038;\\mapsto [x+iy,1-z].<br \/>\n\\end{align*}<br \/>\n\u73b0\u5728, \u5b9a\u4e49Hopf\u7ebf\u4e1b$H$\u5982\u4e0b\uff1a\u4ee4<br \/>\n$$<br \/>\nH=\\set{(w,[z])\\in \\mathbb{C}^2\\times \\mathbb{C}P^1:w=\\lambda z,\\text{ for some $\\lambda\\in\\mathbb{C}^*$}}.<br \/>\n$$<br \/>\n\u82e5\u5b9a\u4e49\u6295\u5c04$\\pi((w,[z]))=[z]$, \u5219\u7ea4\u7ef4$H_{[z]}=\\pi^{-1}([z])=\\set{(\\lambda z,[z]):\\lambda\\in\\mathbb{C}^*}$, \u5176\u5b9e\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u5b9a\u4e49\u5982\u4e0b\uff1a<br \/>\n$$<br \/>\n\\alpha(w,[z])+\\beta(w&#8217;,[z])=(\\alpha w+\\beta w&#8217;, [z]).<br \/>\n$$<br \/>\n\u53ef\u4ee5\u9a8c\u8bc1\u5982\u4e0a\u5b9a\u4e49\u7684$H$\u6ee1\u8db3\u5c40\u90e8\u5e73\u51e1\u5316\u6761\u4ef6, \u4ece\u800c\u662f$\\mathbb{C}P^1$\u4e0a\u7684\u4e00\u4e2a\u590d\u7ebf\u4e1b\u3002<br \/>\n<\/div><br \/>\n\u5c40\u90e8\u5730(\u5728$U_\\alpha$\u4e0a), \u6211\u4eec\u53ef\u4ee5\u9009\u53d6\u7ebf\u4e1b$L$\u5904\u5904\u975e\u96f6\u7684\u622a\u9762$s_a$, \u4f8b\u5982$s_\\alpha(p)=\\phi^{-1}(\\pi(p),1)$.  \u5047\u8bbe$\\xi$\u662f$L$\u7684\u4e00\u4e2a\u6574\u4f53\u622a\u9762, \u5219$\\xi|_{U_\\alpha}=\\xi_\\alpha s_\\alpha$, \u8fd9\u91cc$\\xi_\\alpha\uff1aU_\\alpha\\to\\mathbb{C}$. \u7279\u522b\u5730, \u5728$U_\\alpha\\cap U_\\beta$\u4e0a, $\\xi_\\alpha s_\\alpha=\\xi_\\beta s_\\beta$. \u4e00\u822c\u5730, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u51fd\u6570$g_{\\alpha\\beta}:U_\\alpha\\cap U_\\beta\\to\\mathbb{C}^*$, $s_\\alpha=g_{\\alpha\\beta}s_\\beta$. \u56e0\u6b64, \u5bf9$M$\u7684\u5f00\u8986\u76d6$\\{U_\\alpha\\}$, \u6309\u4e0a\u8ff0\u5b9a\u4e49\u7684$\\{\\xi_\\alpha\\}$\u51b3\u5b9a\u4e86\u4e00\u4e2a<span class=\"latex_em\">\u6574\u4f53\u622a\u9762<\/span>\u5f53\u4e14\u4ec5\u5f53$\\xi_\\beta=g_{\\alpha\\beta}\\xi_\\alpha$. \u53ef\u4ee5\u8bc1\u660e, $L$\u6709\u5904\u5904\u975e\u96f6\u7684\u6574\u4f53\u622a\u9762\u5f53\u4e14\u4ec5\u5f53$L$\u662f\u5e73\u51e1\u7684\u3002 \u4e8b\u5b9e\u4e0a, \u5982\u679c$s$\u662f$L$\u4e0a\u4e00\u4e2a\u5904\u5904\u975e\u96f6\u7684\u6574\u4f53\u622a\u9762, \u5219\u53ef\u5b9a\u4e49\u7ebf\u4e1b\u7684\u540c\u6784$\\Phi: M\\times\\mathbb{C}\\to L$, $(m,\\lambda)\\mapsto \\lambda s(m)$ (\u8bf7\u81ea\u884c\u9a8c\u8bc1\u5b83\u662f\u5e73\u51e1\u7ebf\u4e1b\u5230$L$\u7684\u5fae\u5206\u540c\u80da\u4e14\u9650\u5236\u5728\u6bcf\u6761\u7ea4\u7ef4\u4e0a\u662f\u7ebf\u6027\u540c\u6784).<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 2<\/span> (<span class='latex_defn_name'>\u8f6c\u79fb\u51fd\u6570<\/span>)<span class='latex_defn_h'>.<\/span> \u5982\u4e0a\u5b9a\u4e49\u7684\u51fd\u6570$g_{\\alpha\\beta}:U_\\alpha\\cap U_\\beta\\to\\mathbb{C}^*$\u79f0\u4e3a\u7ebf\u4e1b$L$\u5728\u5c40\u90e8\u5e73\u51e1\u5316$\\{U_\\alpha\\}$\u4e0b\u7684<span class=\"latex_em\">\u8f6c\u79fb\u51fd\u6570<\/span>\u3002<br \/>\n<\/div><br \/>\n\u5bb9\u6613\u9a8c\u8bc1, \u8f6c\u79fb\u51fd\u6570\u6ee1\u8db3\u5982\u4e0b\u7684cocycle\u6761\u4ef6\uff1a<br \/>\n$$<br \/>\ng_{\\alpha\\alpha}=1,\\,\\forall x\\in U_\\alpha,\\forall U_\\alpha, \\quad\\text{ \u4ee5\u53ca }\\quad g_{\\alpha\\beta}g_{\\beta\\gamma}g_{\\gamma\\alpha}=1, \\forall x\\in U_\\alpha\\cap U_\\beta\\cap U_\\gamma.<br \/>\n$$<br \/>\n\u53cd\u8fc7\u6765, \u5bf9\u4efb\u4f55\u6ee1\u8db3\u4e0a\u8ff0cocycle\u6761\u4ef6\u7684\u8f6c\u79fb\u51fd\u6570\u7c07, \u53ef\u4ee5\u6784\u9020\u4e0e$L$\u540c\u6784\u7684\u7ebf\u4e1b\u3002<br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 2<\/span><span class='latex_examp_h'>.<\/span> \u5b9a\u4e49Hopf\u7ebf\u4e1b\u7684\u622a\u9762$s_i:U_i\\to H$,<br \/>\n$$<br \/>\ns_0([z])=\\Bigl(\\bigl(1,z^1\/z^0\\bigr),[z]\\Bigr),\\quad<br \/>\ns_1([z])=\\Bigl(\\bigl(z^0\/z^1,1\\bigr),[z]\\Bigr)<br \/>\n$$<br \/>\n\u5219\u5176\u8f6c\u79fb\u51fd\u6570\u4e3a$g_{01}([z])=z^1\/z^0$.<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 3<\/span><span class='latex_examp_h'>.<\/span> \u5047\u8bbe$E=(E,\\pi,M)$\u662f\u4e00\u4e2a\u79e9\u4e3a$n$\u7684\u590d\u5411\u91cf\u4e1b, \u5219\u5916\u79ef\u4e1b$\\wedge^nE$\u7684\u8f6c\u79fb\u51fd\u6570\u4e3a$\\det g_{\\alpha\\beta}\\in\\mathbb C^*$\u3002 \u5b83\u4e5f\u79f0\u4e3a<span class=\"latex_em\">\u884c\u5217\u5f0f\u4e1b<\/span>, \u5b83\u4e5f\u662f\u4e00\u4e2a\u7ebf\u4e1b\u3002<br \/>\n<\/div><br \/>\n\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$E$\u7684\u8f6c\u79fb\u51fd\u6570\u4e3a$g_{\\alpha\\beta}\\in\\mathrm{GL}(n,\\mathbb C)$, \u800c$\\{s_{\\alpha,i}\\}_{i=1}^n$\u662f$E$\u7684\u5c40\u90e8\u6807\u67b6\u573a. \u8bb0$\\tilde s_\\alpha=s_{\\alpha,1}\\wedge\\cdots\\wedge s_{\\alpha,n}$, \u5219\u7531\u4e8e$s_{\\alpha,i}=g_{\\alpha\\beta}s_{\\beta,i}$, \u6211\u4eec\u77e5\u9053<br \/>\n$$<br \/>\n\\tilde s_\\alpha=g_{\\alpha\\beta}s_{\\beta,1}\\wedge\\cdots\\wedge g_{\\alpha\\beta}s_{\\beta,n}<br \/>\n=\\det g_{\\alpha\\beta}\\tilde s_\\beta.<br \/>\n$$<br \/>\n<span class=\"latex_section\">2.&#x00A0;\u7ebf\u4e1b\u7684\u8054\u7edc<a id=\"sec:2\"><\/a><\/span>\n\n\u7ebf\u4e1b\u7684\u8054\u7edc\u4e3a\u6211\u4eec\u5bf9\u7ebf\u4e1b\u4e0a\u7684\u5bf9\u8c61\u6c42\u5bfc\u7ed9\u51fa\u4e86\u65b9\u6cd5\u3002<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 3<\/span> (<span class='latex_defn_name'>\u7ebf\u4e1b\u7684\u8054\u7edc<\/span>)<span class='latex_defn_h'>.<\/span> \u7ebf\u4e1b$L$\u7684\u4e00\u4e2a\u8054\u7edc\u662f\u7ebf\u6027\u6620\u5c04\uff1a<br \/>\n$$<br \/>\n\\nabla:\\Gamma(M,L)\\to\\Gamma(M,\\Omega^1(L)),\\quad\\Omega^1(L):=T^*M\\times L,<br \/>\n$$<br \/>\n\u4f7f\u5f97\u5982\u4e0b\u7684<span class=\"latex_em\">Liebniz\u6cd5\u5219<\/span>\u6210\u7acb\uff1a<br \/>\n$$<br \/>\n\\nabla(fs)=df\\times s+f\\nabla s,\\quad\\forall f\\in C^\\infty(M,L),\\quad\\forall s\\in\\Gamma(M,L).<br \/>\n$$<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 4<\/span> (<span class='latex_examp_name'>Hopf\u7ebf\u4e1b\u4e0a\u7684\u8bf1\u5bfc\u8054\u7edc<\/span>)<span class='latex_examp_h'>.<\/span> \u56de\u5fc6, $H\\hookrightarrow \\mathbb{C}^2\\times\\mathbb{C}P^1$, \u56e0\u6b64\u622a\u9762$s\\in\\Gamma(H)$\u53ef\u4ee5\u89c6\u4e3a\u51fd\u6570$s:\\mathbb CP^1\\to\\mathbb C^2$, $s([z])=\\lambda z$, $\\lambda\\in\\mathbb C^*$. \u5b9a\u4e49<br \/>\n$$<br \/>\n\\nabla s=(d s)^\\top,<br \/>\n$$<br \/>\n\u8fd9\u91cc$ds$\u662f\u5230$\\mathbb C^2$\u7684\u51fd\u6570\u7684\u5fae\u5206, \u800c${}^\\top$\u662f$\\mathbb C^2$\u5230$H$\u7684\u6b63\u4ea4\u6295\u5f71\u3002<br \/>\n<\/div><br \/>\n<span class=\"latex_subsection\">2.1.&#x00A0;\u8054\u7edc\u7684\u5c40\u90e8\u8868\u793a<a id=\"sec:2.1\"><\/a><\/span>\n\n\u7531\u8054\u7edc\u7684\u57fa\u672c\u6027\u8d28, \u53ef\u4ee5\u77e5\u9053\u8054\u7edc\u5177\u6709\u5c40\u90e8\u6027, \u4ece\u800c\u6211\u4eec\u53ef\u4ee5\u5728\u5c40\u90e8\u5e73\u51e1\u5316\u4e0b\u5c06\u5176\u5177\u4f53\u8868\u793a\u51fa\u6765\u3002<br \/>\n\u5047\u8bbe$s_\\alpha:U_\\alpha\\to L$\u662f\u7ebf\u4e1b$L$\u7684\u4e00\u4e2a\u5904\u5904\u975e\u96f6\u7684\u5c40\u90e8\u622a\u9762\u3002\u5b9a\u4e49$U_\\alpha$\u4e0a\u76841\u5f62\u5f0f$A_\\alpha$, $\\nabla s_\\alpha=A_\\alpha\\otimes s_\\alpha$. \u6ce8\u610f\u5230, \u5bf9$\\xi\\in\\Gamma(M,L)$, \u5c40\u90e8\u5730, $\\xi|_{U_\\alpha}=\\xi_\\alpha s_\\alpha$, $\\xi_\\alpha:U_\\alpha\\to\\mathbb C$, \u6211\u4eec\u6709<br \/>\n$$<br \/>\n(\\nabla\\xi)|_{U_\\alpha}=d\\xi_\\alpha s_\\alpha+\\xi_\\alpha\\nabla s_\\alpha<br \/>\n=(d\\xi_\\alpha+ A_\\alpha\\xi_\\alpha)s_\\alpha.<br \/>\n$$<br \/>\n\u56e0\u6b64, \u5c40\u90e8\u5730, \u6211\u4eec\u5f80\u5f80\u5c06\u8054\u7edc\u5199\u6210$\\nabla=d+A$\u7684\u5f62\u5f0f, \u8fd9\u91cc$A\\in\\Omega^1(U;\\mathrm{End}\\mathbb C)=\\Omega^1(U,\\mathbb C)$\u3002<br \/>\n<span class=\"latex_subsection\">2.2.&#x00A0;\u8054\u7edc\u7684\u8f6c\u79fb\u5173\u7cfb<a id=\"sec:2.2\"><\/a><\/span>\n\n\u56de\u5fc6, \u6309\u7167\u8f6c\u79fb\u51fd\u6570\u7684\u5b9a\u4e49$s_\\alpha=g_{\\alpha\\beta}s_\\beta$, \u6545<br \/>\n$$<br \/>\nA_\\alpha s_\\alpha=ds_\\alpha=dg_{\\alpha\\beta}s_\\beta+g_{\\alpha\\beta}\\nabla s_\\beta=dg_{\\alpha\\beta}g_{\\alpha\\beta}^{-1}s_\\alpha+g_{\\alpha\\beta}A_\\beta g_{\\alpha\\beta}^{-1}s_\\alpha,<br \/>\n$$<br \/>\n\u8fd9\u8868\u660e<br \/>\n$$<br \/>\nA_\\alpha=g_{\\alpha\\beta}^{-1}A_\\beta g_{\\alpha\\beta}+g_{\\alpha\\beta}^{-1}dg_{\\alpha\\beta}<br \/>\n=A_\\beta+g_{\\alpha\\beta}^{-1}dg_{\\alpha\\beta}.<br \/>\n$$<br \/>\n\u4e8b\u5b9e\u4e0a, \u6ee1\u8db3\u4e0a\u8ff0\u5173\u7cfb\u7684\u4e00\u7c07$\\set{A_\\alpha}$\u552f\u4e00\u51b3\u5b9a\u4e86$L$\u7684\u4e00\u4e2a\u8054\u7edc, \u5c40\u90e8\u5730, \u6211\u4eec\u5b9a\u4e49$\\nabla s_\\alpha=A_\\alpha s_\\alpha$\u5373\u53ef\u3002<br \/>\n<span class=\"latex_subsection\">2.3.&#x00A0;\u8054\u7edc\u8bf1\u5bfc\u7684\u5171\u53d8\u5916\u5fae\u5206<a id=\"sec:2.3\"><\/a><\/span>\n\n\u6211\u4eec\u8bb0$\\Omega^k(L):=\\Gamma(M,\\wedge^kT^*M\\otimes L)$, $\\Omega^k(\\mathrm{End}L):=\\Gamma(M,\\wedge^kT^*M\\otimes \\mathrm{End}L)$.<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 4<\/span> (<span class='latex_defn_name'>\u5171\u53d8\u5916\u5fae\u5206<\/span>)<span class='latex_defn_h'>.<\/span> \u82e5$\\nabla$\u662f$L$\u4e0a\u4e00\u8054\u7edc, \u5219\u5bf9$\\omega\\otimes s\\in\\Omega^k(L)$, \u5b9a\u4e49<br \/>\n$$<br \/>\nD(\\omega\\otimes s)=d\\omega\\otimes s+(-1)^k\\omega\\wedge\\nabla s.<br \/>\n$$<br \/>\n\u5bf9$T\\in\\Omega^k(\\mathrm{End}L)$, \u5b9a\u4e49<br \/>\n$$<br \/>\n(DT)s=D(Ts)-(-1)^kT(\\nabla s).<br \/>\n$$<br \/>\n<\/div><br \/>\n\u5c40\u90e8\u5730, \u6211\u4eec\u77e5\u9053$\\nabla=d+A$, \u6545<br \/>\n\\begin{align*}<br \/>\n(DT)s&#038;=D(Ts)-(-1)^kT\\wedge(\\nabla s)\\\\<br \/>\n&#038;=d(Ts)+A\\wedge Ts-(-1)^kT\\wedge(ds+As)\\\\<br \/>\n&#038;=d(Ts)-(-1)^kT\\wedge ds+(A\\wedge T-(-1)^kT\\wedge A)s\\\\<br \/>\n&#038;=(dT+A\\wedge T-(-1)^kT\\wedge A)s.<br \/>\n\\end{align*}<br \/>\n\u6211\u4eec\u5c06\u5176\u7b80\u8bb0\u4e3a<br \/>\n$$<br \/>\nDT=dT+[A\\wedge T].<br \/>\n$$<br \/>\n<span class=\"latex_section\">3.&#x00A0;\u7ebf\u4e1b\u7684\u66f2\u7387<a id=\"sec:3\"><\/a><\/span>\n\n\u56de\u5fc6, \u7ebf\u4e1b\u7684\u8054\u7edc\u5c40\u90e8\u7531\u6ee1\u8db3\u8f6c\u79fb\u5173\u7cfb$A_\\alpha=A_\\beta+g_{\\alpha\\beta}^{-1}dg_{\\alpha\\beta}$\u7684\u4e00\u7c071\u5f62\u5f0f$\\set{A_\\alpha}$\u51b3\u5b9a\u3002\u6ce8\u610f\u5230<br \/>\n$$<br \/>\ndA_\\alpha=dA_\\beta+d(g_{\\alpha\\beta}^{-1}d g_{\\alpha\\beta})<br \/>\n=dA_\\beta.<br \/>\n$$<br \/>\n\u65452\u5f62\u5f0f$dA_\\alpha$\u662f\u6574\u4f53\u5b9a\u4e49\u7684\u3002<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 5<\/span> (<span class='latex_defn_name'>\u8054\u7edc\u7684\u66f2\u7387<\/span>)<span class='latex_defn_h'>.<\/span> \u6211\u4eec\u79f0\u4e0a\u8ff0\u6574\u4f53\u5b9a\u4e49\u76842\u5f62\u5f0f\u4e3a\u8054\u7edc$\\nabla$\u7684<span class=\"latex_em\">\u66f2\u7387<\/span>, \u8bb0\u4f5c$F_\\nabla$\u6216\u8005$F$\u3002<br \/>\n<\/div><br \/>\n<span class=\"latex_subsection\">3.1.&#x00A0;\u66f2\u7387\u7684\u57fa\u672c\u6027\u8d28<a id=\"sec:3.1\"><\/a><\/span>\n\n<div class='latex_prop'><span class='latex_prop_h'>\u547d\u9898 6<\/span><span class='latex_prop_h'>.<\/span> \u5047\u8bbe$F$\u662f\u8054\u7edc$\\nabla$\u7684\u66f2\u7387, \u5219<br \/>\n<ul><li>$F=D^2=D\\nabla$;<\/li><li>\u7b2c\u4e8cBianchi\u6052\u7b49\u5f0f\uff1a $\\nabla F=0$;<\/li><li>\u82e5$\\nabla&#8217;$\u662f\u53e6\u4e00\u8054\u7edc, \u5219\u5b58\u5728\u6574\u4f531\u5f62\u5f0f$\\eta\\in\\Omega^1(M)$, \u4f7f\u5f97<br \/>\n$$<br \/>\n\\nabla&#8217;=\\nabla+\\eta,\\quad<br \/>\nF_{\\nabla&#8217;}=F_\\nabla+d\\eta.<br \/>\n$$<\/li><\/ul><\/div><br \/>\n<div class='latex_proof'><span class='latex_proof_h'>\u8bc1\u660e<\/span><span class='latex_proof_h'>.<\/span> \u4e8b\u5b9e\u4e0a, \u5c40\u90e8\u5730\u5047\u8bbe$s_\\alpha$\u662f\u7ebf\u4e1b$L$\u7684\u5c40\u90e8\u6807\u67b6, \u5219$L$\u7684\u4efb\u610f\u622a\u9762\u53ef\u8868\u793a\u4e3a$s=f_\\alpha s_\\alpha$, \u8fd9\u91cc$f_\\alpha$\u662f$U_\\alpha\\subset M$\u4e0a\u7684\u590d\u503c\u51fd\u6570\u3002\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u5982\u4e0b\uff1a<br \/>\n\\begin{align*}<br \/>\nD^2s&#038;=D(\\nabla s)=D[(d f_\\alpha+f_\\alpha A_\\alpha)s_\\alpha]\\\\<br \/>\n&#038;=d(df_\\alpha+A_\\alpha f_\\alpha)s_\\alpha+(-1)^1(df_\\alpha+A_\\alpha f_\\alpha)\\wedge\\nabla s_\\alpha\\\\<br \/>\n&#038;=[dA_\\alpha f_\\alpha-A_\\alpha\\wedge df_\\alpha-(df_\\alpha+A_\\alpha f_\\alpha)\\wedge A_\\alpha]s_\\alpha\\\\<br \/>\n&#038;=[dA_\\alpha f_\\alpha-A_\\alpha\\wedge df_\\alpha+A_\\alpha\\wedge(df_\\alpha+A_\\alpha f_\\alpha)]s_\\alpha\\\\<br \/>\n&#038;=[dA_\\alpha+A_\\alpha\\wedge A_\\alpha]f_\\alpha s_\\alpha\\\\<br \/>\n&#038;=(dA_\\alpha+A_\\alpha\\wedge A_\\alpha)s.<br \/>\n\\end{align*}<br \/>\n\u4f46\u662f\u6ce8\u610f\u5230, $A_\\alpha$\u662f$U_\\alpha$\u4e0a\u590d\u503c\u76841-\u5f62\u5f0f, \u6545$A_\\alpha\\wedge A_\\alpha=0$. \u4ece\u800c\u6211\u4eec\u5f97\u5230<br \/>\n$$<br \/>\n[D^2 s]|_{U_\\alpha}=dA_\\alpha s=[F_\\nabla s]|_{U_\\alpha}.<br \/>\n$$<\/p>\n<p>\u5bf9\u7b2c\u4e8c\u6761, \u6309\u7167\u5b9a\u4e49\uff1a<br \/>\n\\begin{align*}<br \/>\n\\nabla F|_{U_\\alpha}&#038;=dF|_{U_\\alpha}+[A_\\alpha\\wedge F|_{U_\\alpha}]\\\\<br \/>\n&#038;=d(dA_\\alpha)+[A_\\alpha\\wedge (dA_\\alpha)]\\\\<br \/>\n&#038;=A_\\alpha\\wedge d A_\\alpha-(-1)^2dA_\\alpha\\wedge A_\\alpha\\\\<br \/>\n&#038;=0.<br \/>\n\\end{align*}<\/p>\n<p>\u5bf9\u7b2c\u4e09\u6761, \u5c40\u90e8\u5730, \u6211\u4eec\u77e5\u9053$\\nabla|_{U_\\alpha}=d+A_\\alpha$, $\\nabla&#8217;|_{U_\\alpha}=d+A_\\alpha&#8217;$, \u5047\u8bbe$A_\\alpha&#8217;-A_\\alpha=\\eta_\\alpha\\in \\Gamma(T^*U_\\alpha)$, \u5219<br \/>\n$$<br \/>\n\\eta_\\beta=A_\\beta&#8217;-A_\\beta=A_\\alpha&#8217;-g_{\\alpha\\beta}^{-1}dg_{\\alpha\\beta}-(A_\\alpha-g_{\\alpha\\beta}^{-1}dg_{\\alpha\\beta})<br \/>\n=A_\\alpha&#8217;-A_\\alpha=\\eta_\\alpha.<br \/>\n$$<br \/>\n\u6545$\\eta$\u662f\u6574\u4f53\u76841\u5f62\u5f0f\u3002<\/p>\n<p>\u6700\u540e<br \/>\n$$<br \/>\nF_{\\nabla&#8217;}|_{U_\\alpha}=dA_\\alpha&#8217;=d(A_\\alpha+\\eta_\\alpha)=F_\\nabla|_{U_\\alpha}+d\\eta_\\alpha,<br \/>\n$$<br \/>\n\u56e0\u6b64<br \/>\n$$<br \/>\nF_{\\nabla&#8217;}=F_\\nabla+d\\eta.<br \/>\n$$<br \/>\n<\/div><br \/>\n<span class=\"latex_section\">4.&#x00A0;Hermitian\u7ebf\u4e1b<a id=\"sec:4\"><\/a><\/span>\n\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 7<\/span> (<span class='latex_defn_name'>Hermitian\u7ebf\u4e1b<\/span>)<span class='latex_defn_h'>.<\/span> \u5982\u679c\u7ebf\u4e1b$L$\u4e0a\u5177\u6709\u4e00\u4e2aHermitian\u5185\u79ef$h$, \u5373$h|_m(\\cdot,\\cdot)$\u5b9a\u4e49\u4e86$L_{m}$\u4e0a\u4e00\u4e2aHermitian\u5185\u79ef(\u7b2c\u4e8c\u5206\u91cf\u5171\u8f6d\u7ebf\u6027)\u4e14$h$\u5173\u4e8e$m$\u5149\u6ed1, \u5219\u6211\u4eec\u79f0$(L,h)$\u4e3a\u4e00\u4e2a<span class=\"latex_em\">Hermitian\u7ebf\u4e1b<\/span>\u3002<br \/>\n<\/div><br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 8<\/span> (<span class='latex_defn_name'>\u5168\u7eaf\u7ebf\u4e1b\u53ca\u5176\u622a\u9762<\/span>)<span class='latex_defn_h'>.<\/span> \u5047\u8bbe$L$, $M$\u662f\u4e24\u4e2a<span class=\"latex_em\">\u590d\u6d41\u5f62<\/span>, \u7ebf\u4e1b$(L,\\pi,M)$\u79f0\u4e3a\u662f\u4e00\u4e2a<span class=\"latex_em\">\u5168\u7eaf\u7ebf\u4e1b<\/span>, \u5982\u679c$\\pi:L\\to M$\u662f\u5168\u7eaf\u7684, \u4e14\u8f6c\u79fb\u51fd\u6570$g_{\\alpha\\beta}:U_\\alpha\\cap U_\\beta\\to C^*$\u662f\u5168\u7eaf\u51fd\u6570\u3002<\/p>\n<p>\u5168\u7eaf\u7ebf\u4e1b\u7684\u4e00\u4e2a\u5c40\u90e8\u622a\u9762$s$\u79f0\u4e3a\u662f\u5c40\u90e8\u5168\u7eaf\u622a\u9762, \u5982\u679c\u5728\u5c40\u90e8\u5e73\u51e1\u5316\u4e0b, \u5b83\u662f\u4e00\u4e2a\u5168\u7eaf\u6620\u5c04$s:U_\\alpha\\to U_\\alpha\\times\\mathbb C$. \u660e\u663e\u5730, $s_\\alpha(p)=\\phi^{-1}(\\pi(p),1)$\u5c31\u662f\u4e00\u4e2a\u5c40\u90e8\u5168\u7eaf\u622a\u9762\u3002<\/p>\n<p>\u6211\u4eec\u79f0\u5168\u7eaf\u7ebf\u4e1b\u7684\u4e00\u4e2a\u622a\u9762$s$\u662f<span class=\"latex_em\">\u5168\u7eaf\u622a\u9762<\/span>, \u5982\u679c\u5b83\u5728\u5168\u7eaf\u6807\u67b6\u4e0b\u7684\u8868\u793a\u7cfb\u6570\u662f\u5168\u7eaf\u51fd\u6570\u3002<br \/>\n<\/div><br \/>\n\u56de\u5fc6, \u5bf9\u590d\u6d41\u5f62$M$, \u5176\u5b9e\u6709\u590d\u5750\u6807\u7cfb$z=(z^1,\\ldots,z^n)$. \u5bf9$M$\u4e0a\u7684$(p,q)$\u5f62\u5f0f$\\omega$. \u5728\u5c40\u90e8\u5750\u6807\u7cfb\u4e0b\u53ef\u5c06\u5176\u8868\u793a\u4e3a, $\\omega=f_{IJ}dz^I\\wedge d\\bar z^J$, $|I|=p$, $|J|=q$. \u6545\u5916\u5fae\u5206\u53ef\u5206\u89e3\u4e3a<br \/>\n\\begin{align*}<br \/>\nd\\omega&#038;=df_{IJ}dz^I\\wedge d\\bar z^J\\\\<br \/>\n&#038;=\\partial_{z^l}f_{IJ}dz^l\\wedge dz^I\\wedge d\\bar z^J+\\partial_{\\bar z^l}f_{IJ} d\\bar z^l\\wedge dz^I\\wedge d\\bar z^J\\\\<br \/>\n&#038;:=\\partial\\omega+\\bar\\partial\\omega\\in\\Omega^{p+1,q}\\otimes\\Omega^{(p,q+1)}.<br \/>\n\\end{align*}<br \/>\n\u4e00\u822c\u5730, \u5c06\u4e0a\u5f0f\u7b80\u8bb0\u4e3a<br \/>\n$$<br \/>\nd=\\partial+\\bar\\partial.<br \/>\n$$<\/p>\n<p>\u6ce8\u610f\u5230<br \/>\n$$<br \/>\n0=d^2=(\\partial+\\bar\\partial)^2=\\partial^2<br \/>\n+(\\partial\\bar\\partial+\\bar\\partial\\partial)<br \/>\n+\\bar\\partial^2\\in<br \/>\n\\Omega^{(p+2,q)}\\oplus\\Omega^{(p+1,q+1)}\\oplus\\Omega^{(p,q+2)},<br \/>\n$$<br \/>\n\u6545<br \/>\n$$<br \/>\n\\partial^2=0=\\bar\\partial^2=\\partial\\bar\\partial+\\bar\\partial\\partial.<br \/>\n$$<br \/>\n\u6211\u4eec\u5c06\u770b\u5230, $\\bar\\partial$\u5728\u5168\u7eaf\u7ebf\u4e1b\u4e5f\u53ef\u5b9a\u4e49\u3002<\/p>\n<p>\u82e5$\\nabla$\u662f\u5168\u7eaf\u7ebf\u4e1b$L$\u4e0a\u4e00\u4e2a\u8054\u7edc, $\\Theta$\u662f$L$\u7684\u4e00\u4e2a\u622a\u9762\u3002 \u5c40\u90e8\u5730$\\Theta|_{U_\\alpha}=\\theta_\\alpha\\otimes s_\\alpha$, $\\nabla|_{U_\\alpha}=d+A_\\alpha$, $A_\\alpha$\u662f\u4e00\u4e2a\u590d\u51fd\u6570\u503c\u76841\u5f62\u5f0f\u3002\u5219<br \/>\n\\begin{align*}<br \/>\n\\nabla\\Theta&#038;=(d\\theta_\\alpha+A_\\alpha \\theta_\\alpha)\\otimes s_\\alpha\\\\<br \/>\n&#038;=(\\partial\\theta_\\alpha+A_\\alpha^{(1,0)}\\theta_\\alpha)\\otimes s_\\alpha\\\\<br \/>\n&#038;\\qquad(\\bar\\partial\\theta_\\alpha+A_\\alpha^{(0,1)}\\theta_\\alpha)\\otimes s_\\alpha\\\\<br \/>\n&#038;:=\\nabla&#8217;\\Theta+\\nabla^{\\prime\\prime}\\Theta.<br \/>\n\\end{align*}<br \/>\n\u6ce8\u610f\u5230, $\\nabla=\\nabla&#8217;+\\nabla^{\\prime\\prime}$, \u5b8c\u5168\u4eff\u7167$d=\\partial +\\bar\\partial$. \u53ef\u4ee5\u77e5\u9053<br \/>\n$$<br \/>\n\\nabla^{&#8216;2}=0=\\nabla^{\\prime\\prime 2}<br \/>\n=\\nabla&#8217;\\nabla^{\\prime\\prime}+\\nabla^{\\prime\\prime}\\nabla&#8217;.<br \/>\n$$<\/p>\n<p>\u73b0\u5728, \u5047\u8bbe$L=(L,\\pi,M)$\u662f\u4e00\u4e2a\u5168\u7eaf\u7ebf\u4e1b, $s_\\alpha$\u662f\u5176\u5c40\u90e8\u5168\u7eaf\u622a\u9762\u3002\u5047\u8bbe$\\Theta\\in\\Omega^{p,q}(L)$\u662f$L$\u7684\u4e00\u4e2a$(p,q)$\u5f62\u5f0f\u7684\u622a\u9762, \u5c40\u90e8\u5730, $\\Theta_\\alpha:=\\Theta|_{U_\\alpha}=\\theta_\\alpha \\otimes s_\\alpha$, $\\theta_\\alpha\\in \\Omega^{(p,q)}(U_\\alpha):=\\wedge^{(p,q)}T^*U_\\alpha$. \u5b9a\u4e49<br \/>\n$$<br \/>\n\\bar\\partial\\Theta_\\alpha:=\\bar\\partial\\Theta|_{U_\\alpha}:=\\bar\\partial\\theta_\\alpha\\otimes s_\\alpha\\in\\Omega^{(p,q+1)}(L|_{U_\\alpha}).<br \/>\n$$<br \/>\n\u6ce8\u610f\u5230$s_\\alpha=g_{\\alpha\\beta}s_\\beta$, \u6545<br \/>\n$$<br \/>\n\\bar\\partial\\Theta_\\alpha=\\bar\\partial\\theta_\\alpha\\otimes s_\\alpha<br \/>\n=\\bar\\partial(\\theta_\\alpha)g_{\\alpha\\beta}s_\\beta<br \/>\n=\\bar\\partial(g_{\\alpha\\beta}\\theta_\\alpha)s_\\beta<br \/>\n=\\bar\\partial\\theta_\\beta s_\\beta<br \/>\n=\\bar\\partial\\Theta_\\beta,<br \/>\n$$<br \/>\n\u8fd9\u91cc, \u6211\u4eec\u7528\u5230\u4e86$g_{\\alpha\\beta}$\u662f\u5168\u7eaf\u7684\u3002 \u4e0a\u8ff0\u8ba1\u7b97\u8868\u660e$\\bar\\partial\\Theta$\u662f\u6574\u4f53\u5b9a\u4e49\u7684\u3002<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 9<\/span> (<span class='latex_defn_name'>\u5168\u7eaf\u7ebf\u4e1b\u7684$\\bar\\partial$-\u7b97\u5b50<\/span>)<span class='latex_defn_h'>.<\/span> \u4e0a\u8ff0\u5b9a\u4e49\u7684\u7b97\u5b50$\\bar\\partial:\\Omega^{(p,q)}(L)\\to\\Omega^{(p,q+1)}(L)$\u79f0\u4e3a\u5168\u7eaf\u7ebf\u4e1b\u7684$\\bar\\partial$\u7b97\u5b50\u3002\u4e3a\u4e86\u5f3a\u8c03\u7ebf\u4e1b, \u6709\u65f6\u4e5f\u8bb0\u4e3a$\\bar\\partial_L$.<br \/>\n<\/div><br \/>\n<div class='latex_prop'><span class='latex_prop_h'>\u547d\u9898 10<\/span> (<span class='latex_prop_name'>$\\bar\\partial_L^2=0$<\/span>)<span class='latex_prop_h'>.<\/span> \u5047\u8bbe$\\bar\\partial_L$\u662f\u5982\u4e0a\u5b9a\u4e49\u7684\u5168\u7eaf\u7ebf\u4e1b$L$\u4e0a\u7684\u7b97\u5b50, \u5219<br \/>\n$$<br \/>\n\\bar\\partial_L^2=0.<br \/>\n$$<br \/>\n<\/div><br \/>\n<div class='latex_proof'><span class='latex_proof_h'>\u8bc1\u660e<\/span><span class='latex_proof_h'>.<\/span> \u7531$\\bar\\partial_L$\u7684\u5c40\u90e8\u5b9a\u4e49\u4ee5\u53ca$\\bar\\partial^2=0$\u5373\u5f97\u3002<br \/>\n<\/div><br \/>\n\u5728\u5177\u6709Hermitian\u5ea6\u91cf\u7684\u5168\u7eaf\u7ebf\u4e1b\u4e0a, \u53ef\u4ee5\u9009\u62e9\u4e00\u4e2a\u5178\u5219\u8054\u7edc(\u79f0\u4e3a<span class=\"latex_em\">Chern\u8054\u7edc<\/span>).<br \/>\n<div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 11<\/span> (<span class='latex_thm_name'>Chern\u8054\u7edc\u7684\u5b58\u5728\u552f\u4e00\u6027<\/span>)<span class='latex_thm_h'>.<\/span> \u5047\u8bbe$(L,\\pi,M)$\u662f\u4e00\u4e2a\u5168\u7eaf\u7ebf\u4e1b\u800c$h$\u662f$L$\u4e0a\u7684Hermitian\u5ea6\u91cf(\u5c40\u90e8\u53ef\u7531$U_\\alpha\\times\\mathbb C\\subset\\mathbb C^{\\dim M+1}$\u4e0a\u7684Hermitian\u5185\u79ef\u7ed9\u51fa\u3002\u5229\u7528\u5355\u4f4d\u5206\u89e3\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6574\u4f53\u5185\u79ef)\u3002\u5219\u5b58\u5728$L$\u7684\u552f\u4e00\u4e00\u4e2a\u8054\u7edc$\\nabla$\u4f7f\u5f97<br \/>\n<ul><li>$\\nabla$\u548c$h$\u76f8\u5bb9\u3002 \u5373\u5bf9$L$\u7684\u4efb\u4f55\u4e24\u4e2a\u622a\u9762$s_1,s_2$, \u6211\u4eec\u6709<br \/>\n$$<br \/>\nd\\inner{s_1,s_2}_h=\\inner{\\nabla s_1,s_2}_h+\\inner{s_1,\\nabla s_2}_h.<br \/>\n$$<\/li><li>\u548c\u5168\u7eaf\u7ed3\u6784\u76f8\u5bb9\u3002\u5373<br \/>\n$$<br \/>\n\\nabla^{\\prime\\prime}=\\bar\\partial_L.<br \/>\n$$<\/li><\/ul><\/div><br \/>\n<div class='latex_proof'><span class='latex_proof_h'>\u8bc1\u660e<\/span><span class='latex_proof_h'>.<\/span> \u7531\u4e8e$L$\u662f\u4e00\u4e2a\u7ebf\u4e1b, \u6545\u5c40\u90e8\u622a\u9762$s_\\alpha$\u5373\u4e3a\u5176\u5c40\u90e8\u6807\u67b6\u3002 $h_\\alpha:=h(s_\\alpha,s_\\alpha)$. \u5219\u7531\u5ea6\u91cf\u76f8\u5bb9\u6027\uff1a<br \/>\n$$<br \/>\n\\partial h_\\alpha+\\bar\\partial h_\\alpha=dh_\\alpha=A_\\alpha h_\\alpha+\\bar h_\\alpha A_\\alpha.<br \/>\n$$<br \/>\n\u800c\u7531\u548c\u5168\u7eaf\u7ed3\u6784\u7684\u76f8\u5bb9\u6027\u77e5\uff1a<br \/>\n$$<br \/>\n(A_\\alpha^{(1,0)}+A_\\alpha^{(0,1)})s_\\alpha=A_\\alpha s_\\alpha=\\nabla s_\\alpha=\\nabla&#8217; s_\\alpha+\\bar\\partial_L s_\\alpha=\\nabla&#8217;s_\\alpha=A_\\alpha^{(1,0)}s_\\alpha,<br \/>\n$$<br \/>\n\u6545$A_\\alpha^{(0,1)}=0$\u4e14$A_\\alpha=A_\\alpha^{(1,0)}$\u3002<\/p>\n<p>\u4e8e\u662f\u6211\u4eec\u5f97\u5230<br \/>\n$$<br \/>\n\\partial h_\\alpha+\\bar\\partial h_\\alpha=A_\\alpha^{(1,0)}h_\\alpha+h_\\alpha \\overline{A_\\alpha^{(1,0)}}\\implies \\partial h_\\alpha =A_\\alpha h_\\alpha\\in \\Omega^{(1,0)}(U_\\alpha)<br \/>\n$$<br \/>\n\u5373<br \/>\n$$<br \/>\n\\partial\\ln h_\\alpha=A_\\alpha,\\quad\\bar\\partial h_\\alpha=\\bar A_\\alpha.<br \/>\n$$<br \/>\n\u6ce8\u610f, \u7ebf\u4e1b$A_\\alpha$\u662f$U_\\alpha$\u4e0a\u67d0\u4e2a\u590d\u51fd\u6570\u503c\u76841\u5f62\u5f0f, \u800c\u7531\u6b63\u5b9a\u6027, $h_\\alpha$\u662f$U_\\alpha$\u67d0\u4e2a\u5927\u4e8e\u96f6\u7684\u5b9e\u503c\u51fd\u6570\u3002\u6545$\\bar\\partial \\ln h_\\alpha =\\overline{\\partial\\ln h_\\alpha}=\\bar A_\\alpha$, \u5373\u53ea\u6709\u4e00\u4e2a\u65b9\u7a0b\uff1a<br \/>\n$$<br \/>\nA_\\alpha=\\partial\\ln h_\\alpha.<br \/>\n$$<br \/>\n\u53ef\u89c1, \u5728\u5b9a\u7406\u7ed9\u5b9a\u7684\u4e24\u4e2a\u6761\u4ef6\u4e0b, \u8054\u7edc\u552f\u4e00\u7531\u5ea6\u91cf\u51b3\u5b9a\u3002\u53cd\u8fc7\u6765, \u53ef\u4ee5\u9a8c\u8bc1, \u5c40\u90e8\u5982\u4e0a\u5b9a\u4e49\u7684\u8054\u7edc\u6ee1\u8db3\u5b9a\u7406\u7684\u4e24\u4e2a\u6761\u4ef6\u3002\u8fd9\u5c31\u5b8c\u6210\u4e86\u8bc1\u660e\u3002<br \/>\n<\/div><br \/>\n<span class=\"latex_subsection\">4.1.&#x00A0;Hermitian\u5168\u7eaf\u7ebf\u4e1b\u7684\u66f2\u7387<a id=\"sec:4.1\"><\/a><\/span>\n\n\u5047\u8bbe$\\nabla$\u662fHermitian\u5168\u7eaf\u7ebf\u4e1b$L$\u7684Chern\u8054\u7edc\u3002\u7531\u7ebf\u4e1b\u66f2\u7387\u7684\u5c40\u90e8\u8868\u793a\uff1a<br \/>\n$$<br \/>\nF_\\nabla=dA_\\alpha=(\\partial+\\bar\\partial)\\partial\\ln h_\\alpha<br \/>\n=\\bar\\partial\\partial\\ln h_\\alpha<br \/>\n=-\\partial\\bar\\partial\\ln h_\\alpha.<br \/>\n$$<\/p>\n<p>\u56de\u5fc6, \u590d\u7ebf\u4e1b$L$\u4e0a\u7684\u590d\u7ed3\u6784$J=i=\\begin{pmatrix}0&#038;-1\\\\1&#038;0\\end{pmatrix}$(\u89c6\u4e3a\u5b9e2\u7ef4\u7a7a\u95f4\u7684\u590d\u5316), \u6545\u6211\u4eec\u53ef\u5c06$L$\u7684\u622a\u9762\u6309\u7167$J$\u7684\u7279\u5f81\u7a7a\u95f4\u5206\u89e3\u4e3a$\\Gamma(L)=\\Omega^0{L}=\\Omega^+(L)\\oplus\\Omega^-(L):=i\\Omega(L)\\oplus(-i)\\Omega(L)$. \u8fdb\u800c\u7c7b\u4f3c\u5f97\u5230$\\Omega^k(L)=\\oplus_{i=0^k}\\Omega^{(i,k-i)}(L)$. \u7279\u522b\u5730, \u5bf9Chern\u8054\u7edc, \u6211\u4eec\u6709$A_\\alpha^{(0,1)}=0$. \u5373$A_\\alpha=A_\\alpha^{(1,0}\\in i\\Omega^1$. <\/p>\n<p>\u73b0\u5728\u7531\u4e8e$h_\\alpha$\u662f\u5927\u4e8e\u96f6\u7684\u5b9e\u51fd\u6570, \u5229\u7528Chern\u8054\u7edc\u7684\u5ea6\u91cf\u76f8\u5bb9\u6027,<br \/>\n\\begin{align*}<br \/>\n0&#038;=d^2\\inner{s_\\alpha,s_\\alpha}=d(\\inner{\\nabla s_\\alpha,s_\\alpha}+\\inner{s_\\alpha,\\nabla s_\\alpha})\\\\<br \/>\n&#038;=\\inner{D^2s_\\alpha,s_\\alpha}-\\inner{\\nabla s_\\alpha,\\nabla s_\\alpha}+\\inner{\\nabla s_\\alpha,\\nabla s_\\alpha}+\\inner{s_\\alpha,D^2 s_\\alpha}\\\\<br \/>\n&#038;=\\inner{D^2s_\\alpha,s_\\alpha}+\\inner{s_\\alpha,D^2 s_\\alpha}\\\\<br \/>\n&#038;=F_\\nabla h_\\alpha+\\bar F_\\nabla h_\\alpha.<br \/>\n\\end{align*}<br \/>\n\u6545\u6211\u4eec\u5f97\u5230<br \/>\n$$<br \/>\n\\bar F=-F,<br \/>\n$$<br \/>\n\u5373\u5bf9Chern\u8054\u7edc, $F_\\nabla$\u662f\u7eaf\u865a\u6570\u503c\u7684(1,1)-\u5f62\u5f0f\u3002<br \/>\n<span class=\"latex_section\">5.&#x00A0;\u7ebf\u4e1b\u7684\u548c\u4e50\u7fa4<a id=\"sec:5\"><\/a><\/span>\n\n\u5047\u8bbe$\\gamma(t):[0,1]\\to M$, $p=\\gamma(0)$, $q=\\gamma(1)$, \u662f\u4e00\u6761\u8fde\u63a5$p$,$q$\u7684\u5149\u6ed1\u66f2\u7ebf, $\\xi_0\\in T_pL$\u3002<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 12<\/span> (<span class='latex_defn_name'>\u5e73\u884c\u79fb\u52a8<\/span>)<span class='latex_defn_h'>.<\/span> \u5b9a\u4e49$\\xi_0$\u6cbf\u7740\u66f2\u7ebf$\\gamma$\u5173\u4e8e\u8054\u7edc$\\nabla$\u5e73\u884c\u79fb\u52a8\u5230$q$\u7684\u5411\u91cf\u4e3a$\\xi_1$, \u8fd9\u91cc$\\xi_1=\\xi(1)$, \u5176\u4e2d$\\xi(t)$\u662f\u6cbf\u7740$\\gamma$\u7684<span class=\"latex_em\">\u5e73\u79fb\u5411\u91cf\u573a<\/span>, \u5373\u6ee1\u8db3$\\xi(0)=\\xi_0$\u4e14<br \/>\n$$<br \/>\n\\nabla_{\\dot\\gamma}\\xi(t)=0.<br \/>\n$$<br \/>\n<\/div><br \/>\n\u53ef\u4ee5\u8bc1\u660e\u4e0a\u8ff0\u5e73\u884c\u79fb\u52a8\u5b9a\u4e49\u7684$P_\\gamma: T_pL\\to T_qL$\u662f\u4e00\u4e2a\u7ebf\u6027\u540c\u6784\u3002<br \/>\n\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$\\xi(t)=\\xi_\\alpha(t)s_\\alpha(\\gamma(t))$, \u5219<br \/>\n$$<br \/>\n0=\\nabla_{\\dot\\gamma}\\xi(t)=\\left(\\frac{d\\xi^\\alpha(t)}{dt}+\\xi_\\alpha(t)\\dot\\gamma\\cdot A_\\alpha(\\gamma)\\right)s_\\alpha\\circ\\gamma(t).<br \/>\n$$<br \/>\n\u56e0\u6b64, \u6211\u4eec\u5f97\u5230\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4<br \/>\n$$<br \/>\n\\frac{d(\\xi_\\alpha(t))}{dt}=-\\dot\\gamma\\cdot A_\\alpha(\\gamma)\\xi_\\alpha.<br \/>\n$$<br \/>\n\u8fd9\u91cc$\\dot\\gamma\\cdot A_\\alpha$\u8868\u793a1\u5f62\u5f0f\u4e0e\u5207\u5411\u91cf\u573a\u7684\u7f29\u5e76\u3002\u7531\u5e38\u5fae\u5206\u65b9\u7a0b\u7ec4\u7684\u5b58\u5728\u552f\u4e00\u6027\u5b9a\u7406, \u6211\u4eec\u77e5\u9053,<br \/>\n$$<br \/>\n\\xi_\\alpha(\\gamma(t))=\\exp\\left(-\\int_0^t\\dot\\gamma\\cdot A_\\alpha\\circ\\gamma\\right)\\xi_\\alpha(\\gamma(0)).<br \/>\n$$<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 13<\/span> (<span class='latex_defn_name'>\u548c\u4e50\u7fa4<\/span>)<span class='latex_defn_h'>.<\/span> \u5b9a\u4e49\u7ebf\u4e1b$L$\u7684\u8054\u7edc$\\nabla$\u6cbf\u7740<span class=\"latex_em\">\u95ed\u66f2\u7ebf<\/span>$\\gamma$\u7684<span class=\"latex_em\">\u548c\u4e50\u7fa4<\/span>$\\mathrm{hol}(\\gamma,\\nabla)$\u4e3a<br \/>\n$$<br \/>\nP_\\gamma(s)=\\mathrm{hol}(\\gamma,\\nabla) s,<br \/>\n$$<br \/>\n\u5176\u4e2d$s\\in T_{\\gamma(0)}L$.<br \/>\n<\/div><br \/>\n\u7531\u4e8e\u5e73\u884c\u79fb\u52a8\u4fdd\u6301\u5411\u91cf\u7684\u957f\u5ea6, \u6545\u82e5$L$\u5177\u6709Hermitian\u5ea6\u91cf, \u5219$\\mathrm{hol}(\\gamma,\\nabla)$\u662f\u9149\u7fa4$U(1)$\u7684\u5b50\u7fa4\u3002<\/p>\n<p>\u7531Stocks\u5b9a\u7406, \u6211\u4eec\u77e5\u9053\u5047\u8bbe\u95ed\u66f2\u7ebf$\\gamma$\u56f4\u6210\u76842\u7ef4\u5b50\u6d41\u5f62\u4e3a$\\Sigma$, \u5e76\u4ee4$p=\\gamma(0)=\\gamma(1)$, $\\vec A_\\alpha=(A_1,A_2)$, $A_\\alpha=\\sum_{i=1}^2A_idx^i$, \u5219<br \/>\n\\begin{align*}<br \/>\n\\mathrm{hol}(\\gamma,\\nabla)s&#038;=P_\\gamma(s)=\\xi_\\alpha(\\gamma(1))s_\\alpha(\\gamma(1))\\\\<br \/>\n&#038;=\\exp\\left(-\\int_0^1\\dot\\gamma\\cdot A_\\alpha(\\gamma)dt\\right)\\xi_\\alpha(\\gamma(0))s_\\alpha(\\gamma(0))\\\\<br \/>\n&#038;=\\exp\\left(-\\int_\\gamma \\vec A_\\alpha(\\gamma)d\\gamma\\right)s\\\\<br \/>\n&#038;=\\exp\\left(\\int_\\Sigma \\mathrm{div}\\vec A_\\alpha^\\perp dS\\right)s\\\\<br \/>\n&#038;=\\exp\\left(-\\int_\\Sigma dA_\\alpha dS\\right)s\\\\<br \/>\n&#038;=\\exp\\left(-\\int_\\Sigma F_\\nabla\\right)s.<br \/>\n\\end{align*}<br \/>\n\u8fd9\u91cc, \u7b2c3\u884c\u7528\u5230\u4e86<a href=\"https:\/\/en.wikipedia.org\/wiki\/Line_integral#Definition_2\">\u7ebf\u79ef\u5206\u4e0e\u9053\u8def\u79ef\u5206\u7684\u5b9a\u4e49<\/a>. \u7b2c4\u884c\u7528\u5230\u4e86<a href=\"https:\/\/en.wikipedia.org\/wiki\/Stokes%27_theorem#Kelvin.E2.80.93Stokes_theorem\">\u6563\u5ea6\u5b9a\u7406<\/a>, \u5176\u4e2d$\\vec A_\\alpha^\\perp=(-A_2,A_1)$. \u6545$\\mathrm{div}\\vec A_\\alpha^\\perp=-\\partial_1A_2+\\partial_2A_1$. \u6ce8\u610f\u5230$dA_\\alpha=(\\partial_1A_2-\\partial_2A_1)dx^1\\wedge dx^2=-\\mathrm{div}\\vec A_\\alpha^\\perp dS$. \u6700\u540e\u4e00\u4e2a\u7b49\u5f0f\u7528\u5230\u66f2\u7387\u7684\u5c40\u90e8\u8ba1\u7b97\u516c\u5f0f\u3002<br \/>\n<div class='latex_rmk'><span class='latex_rmk_h'>\u6ce8\u8bb0 2<\/span><span class='latex_rmk_h'>.<\/span> \u8fd9\u91cc$\\Sigma$\u7684\u5b9a\u4e49\u5e94\u8be5\u7406\u89e3\u4e3a\uff1a\u5bf9\u67d0\u4e2a\u8fde\u7eed\u6620\u5c04$\\Gamma:\\Delta_2\\to M$\u7684\u50cf, \u8fd9\u91cc\u8981\u6c42\u8be5\u6620\u5c04\u6ee1\u8db3$\\Gamma|_{\\partial\\Delta_2}=\\gamma$. \u5373$\\Sigma=\\Gamma(\\Delta_2)$. \u56de\u5fc6$\\Delta_2$\u662f\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u7684\u6807\u51c62\u5355\u5f62\u3002<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 5<\/span><span class='latex_examp_h'>.<\/span> \u8ba1\u7b97\u6807\u51c6\u7403\u9762$S^2$\u4e0aLevi-Civita\u8054\u7edc\u6cbf\u7740\u4e0b\u56fe\u6240\u793a\u7684\u95ed\u8def\u5f97\u5230\u7684\u548c\u4e50\u7fa4\u3002<br \/>\n<a href=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/12\/holonomy-loop-of-sphere.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/12\/holonomy-loop-of-sphere.png\" alt=\"\" width=\"329\" height=\"339\" class=\"aligncenter size-full wp-image-483\" srcset=\"https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/12\/holonomy-loop-of-sphere.png 329w, https:\/\/blog.vanabel.cn\/wp-content\/uploads\/2017\/12\/holonomy-loop-of-sphere-291x300.png 291w\" sizes=\"auto, (max-width: 329px) 100vw, 329px\" \/><\/a><br \/>\n<\/div><br \/>\n\u53d6\u7403\u9762\u7684\u6781\u5750\u6807$(\\theta,\\phi)\\in[0,2\\pi)\\times[0,\\pi)$, \u5219<br \/>\n$$<br \/>\n\\begin{cases}<br \/>\nx=\\sin\\phi\\cos\\theta\\\\<br \/>\ny=\\sin\\phi\\sin\\theta\\\\<br \/>\nz=\\cos\\phi.<br \/>\n\\end{cases}<br \/>\n$$<br \/>\n\u5bb9\u6613\u5f97\u5230\u5750\u6807\u66f2\u7ebf\u7684\u5207\u5411\u91cf\u5206\u522b\u4e3a<br \/>\n\\begin{align*}<br \/>\n\\partial_\\theta&#038;=-\\sin\\phi\\sin\\theta\\partial_x+\\sin\\phi\\cos\\theta\\partial_y\\\\<br \/>\n\\partial_\\phi&#038;=\\cos\\phi\\cos\\theta\\partial_x+\\cos\\phi\\sin\\theta\\partial_y-\\sin\\phi\\partial_z.<br \/>\n\\end{align*}<br \/>\n\u56e0\u6b64, \u5355\u4f4d\u6cd5\u5411\u91cf\u4e3a<br \/>\n\\begin{align*}<br \/>\n\\vec n&#038;=\\frac{\\partial_\\phi\\times\\partial_\\theta}{|\\partial_\\phi\\times\\partial_\\theta|}\\\\<br \/>\n&#038;=\\sin\\phi\\cos\\theta\\partial_x+\\sin\\phi\\sin\\theta\\partial_y+\\cos\\phi\\partial_z\\\\<br \/>\n&#038;=\\sin\\phi\\left(\\partial_\\phi\\times\\partial_\\theta\\right).<br \/>\n\\end{align*}<br \/>\n\u4e3a\u4e86\u8ba1\u7b97$S^2$\u7684Levi-Civta\u8054\u7edc, \u56de\u5fc6, \u5c40\u90e8\u5730, \u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u5904\u5904\u975e\u96f6\u7684\u5750\u6807\u5207\u5411\u91cf\u573a$s$, \u4f7f\u5f97$\\nabla s=As$. \u8fd9\u91cc$A$\u662f\u4e00\u4e2a\u590d\u6570\u3002 \u9009\u53d6\u7ecf\u7ebf\u7684\u5207\u5411\u91cf\u573a\u4f5c\u4e3a\u84dd\u8272\u533a\u57df\u7684\u5c40\u90e8\u622a\u9762$s$, \u5373<br \/>\n$$<br \/>\ns=\\partial_\\phi,<br \/>\n$$<br \/>\n\u5219\u5c06\u5176\u89c6\u4e3a$\\mathbb R^3$\u4e2d\u7684\u5411\u91cf$s=(\\cos\\phi\\cos\\theta,\\cos\\phi\\sin\\theta,-\\sin\\phi)$. \u4ece\u800c\u666e\u901a\u5916\u5fae\u5206\u5f97\u5230<br \/>\n\\begin{align*}<br \/>\nds&#038;=(-\\sin\\phi\\cos\\theta,-\\sin\\phi\\sin\\theta,-\\cos\\phi)d\\phi\\\\<br \/>\n&#038;\\qquad+(-\\cos\\phi\\sin\\theta,\\cos\\phi\\cos\\theta,0)d\\theta.<br \/>\n\\end{align*}<br \/>\n\u6545\u6309\u7167\u5b50\u6d41\u5f62\u8bf1\u5bfc\u8054\u7edc\u7684\u57fa\u672c\u6027\u8d28\uff1a<br \/>\n\\begin{align*}<br \/>\n\\nabla s&#038;=(ds)^\\top\\\\<br \/>\n&#038;=ds-\\langle ds,\\vec n\\rangle\\vec n\\\\<br \/>\n&#038;=ds+\\vec n\\cdot d\\phi\\\\<br \/>\n&#038;=(-\\cos\\phi\\sin\\theta,\\cos\\phi\\cos\\theta,0)d\\theta\\\\<br \/>\n&#038;=\\cos\\phi(-\\sin\\theta,\\cos\\theta,0)d\\theta.<br \/>\n\\end{align*}<br \/>\n\u56de\u5fc6, \u5982\u4e0a\u6784\u9020\u7684Levi-Civita\u8054\u7edc$\\nabla$\u4fdd\u6301\u5ea6\u91cf, \u6545\u5176\u5c40\u90e8\u8868\u793a$A$\u662f$SO(2)$\u503c\u76841\u5f62\u5f0f\u3002 \u4ece\u800c\u5b83\u662f\u67d0\u4e2a\u65cb\u8f6c, \u8fdb\u800c\u53ef\u4ee5\u8868\u793a\u4e3a$e^{i\\tau}$, \u5bf9\u67d0\u4e2a\u89d2\u5ea6$\\tau\\in[0,2\\pi)$\u3002 \u4e3a\u4e86\u770b\u51fa\u8be5\u65cb\u8f6c, \u6ce8\u610f\u5230\u4e0a\u8ff0\u8ba1\u7b97\u8868\u660e$\\nabla \\partial_\\phi\/\/\\partial_\\theta d\\theta$\u3002 \u6709\u6ce8\u610f\u5230$\\partial_\\phi\\perp\\partial_\\theta$, \u6545(\u6ce8\u610f\u5b9a\u5411)<br \/>\n$$<br \/>\n\\sin\\phi\\cdot\\partial_\\phi=-e^{i\\pi\/2}\\partial_\\theta.<br \/>\n$$<br \/>\n\u4ee3\u5165\u5f97\u5230<br \/>\n$$<br \/>\n\\nabla s=\\cos\\phi\/\\sin\\phi\\partial_\\theta d\\theta<br \/>\n=i\\cos\\phi\\partial_\\phi=i\\cos\\phi d\\theta s.<br \/>\n$$<br \/>\n\u5373$A=i\\cos\\phi d\\theta$. \u6709\u6ce8\u610f\u5230, \u4f53\u79ef\u5143$dv_g=\\sin\\phi d\\phi\\wedge d\\theta$, \u4ece\u800c<br \/>\n$$<br \/>\nF_\\nabla=dA=-i\\sin\\phi d\\phi\\wedge d\\theta=i\\sin\\phi d\\theta\\wedge d\\phi<br \/>\n=-i dv_g.<br \/>\n$$<br \/>\n\u56e0\u6b64, \u6839\u636e\u524d\u9762\u66f2\u7387\u4e0e\u548c\u4e50\u7fa4\u7684\u8ba1\u7b97\u5173\u7cfb,<br \/>\n\\begin{align*}<br \/>\n\\mathrm{hol}(\\nabla,\\gamma)&#038;=\\exp\\left(-\\int_{\\Sigma}F_\\nabla\\right)=\\exp\\left(\\int_{\\Sigma}idv_g\\right)\\\\<br \/>\n&#038;=\\exp\\left(\\int_{\\theta_1}^{\\theta_2}\\int_0^{\\pi\/2}i\\sin\\phi d\\theta d\\phi\\right)\\\\<br \/>\n&#038;=\\exp\\left(-i(\\theta_2-\\theta_1)\\cos\\phi|_0^{\\pi\/2}\\right)\\\\<br \/>\n&#038;=\\exp(i(\\theta_2-\\theta_1)).<br \/>\n\\end{align*}<br \/>\n\u5373\u548c\u4e50\u7fa4\u7684\u4f5c\u7528\u76f8\u5bf9\u4e8e\u5c06\u5207\u5411\u91cf\u4f5c$\\theta_2-\\theta_1$\u7684\u65cb\u8f6c, \u8fd9\u4ece\u56fe\u4e2d\u4e5f\u662f\u5bb9\u6613\u770b\u51fa\u7684\u3002<br \/>\n<span class=\"latex_subsection\">5.1.&#x00A0;\u548c\u4e50\u7fa4\u4e0e\u66f2\u7387\u7684\u5173\u7cfb<a id=\"sec:5.1\"><\/a><\/span>\n\n\u5047\u8bbe$X$, $Y$\u662f$M$\u7684\u4e24\u4e2a\u5207\u5411\u91cf, \u5e76\u8bb0$\\Sigma_t$\u4e3a\u7531$X$,$Y$\u6309\u7167\u6d4b\u5730\u7ebf\u5f20\u6210\u7684\u8fb9\u957f\u4e3a$\\sqrt t$\u7684\u66f2\u8fb9\u5e73\u884c\u56db\u8fb9\u5f62\u3002 \u5219\u6211\u4eec\u6709\u4e0b\u5217Taylor\u5c55\u5f00\uff1a<br \/>\n$$<br \/>\n\\mathrm{hol}(\\nabla,\\Sigma_t)=1+tF(X,Y)+o(t).<br \/>\n$$<br \/>\n\u8bc1\u660e\u53ef\u4ee5\u53c2\u8003<a href=\"http:\/\/www.deaneyang.com\/papers\/holonomy.pdf\">Deane Yang<\/a>.<br \/>\n<span class=\"latex_section\">6.&#x00A0;\u9648\u7c7b<a id=\"sec:6\"><\/a><\/span>\n\n\u5047\u8bbe$\\Sigma$\u662f\u4e00\u4e2a\u95ed\u66f2\u9762, $L$\u662f$\\Sigma$\u4e0a\u7684\u4e00\u4e2a\u7ebf\u4e1b, \u800c$\\nabla$, $\\nabla&#8217;=\\nabla+d\\eta$\u662f\u5176\u4e0a\u4e24\u4e2a\u8054\u7edc, \u5176\u4e2d$\\eta$\u662f\u6574\u4f53\u76841\u5f62\u5f0f\u3002 \u5219$F_{\\nabla&#8217;}=F_\\nabla+d\\eta$, \u4ece\u800c\u7531Stokes\u5b9a\u7406,<\/p>\n<p>$$<br \/>\n\\int_{\\Sigma}F_{\\nabla&#8217;}=\\int_\\Sigma F_\\nabla+d\\eta<br \/>\n=\\int_\\Sigma F_\\nabla+\\int_{\\partial\\Sigma=\\emptyset}\\eta<br \/>\n=\\int_\\Sigma F_\\nabla.<br \/>\n$$<br \/>\n\u8fd9\u8868\u660e, \u5bf9\u95ed\u66f2\u9762, \u5168\u66f2\u7387(\u5373\u66f2\u7387\u5728\u6574\u4e2a\u66f2\u9762\u4e0a\u7684\u79ef\u5206)\u4e0d\u4f9d\u8d56\u4e8e\u8054\u7edc\u7684\u9009\u62e9\u3002<\/p>\n<p>\u4e0b\u9762\u5047\u8bbe$\\Sigma_t$\u662f\u66f2\u9762$\\Sigma$\u4e0a\u4e00\u5217\u534a\u5f84\u4e3a$t$\u7684\u6d4b\u5730\u5706\u76d8, \u5e76\u8bb0$\\Sigma^c_t=\\Sigma\\setminus\\Sigma_t$, $\\gamma=\\Sigma_t^c\\cap\\Sigma_t$. \u5219\u6309\u7167\u548c\u4e50\u7fa4\u4e0e\u66f2\u7387\u7684\u5173\u7cfb, \u6211\u4eec\u6709(\u6ce8\u610f\u5b9a\u5411)<br \/>\n$$<br \/>\n\\mathrm{hol}(\\nabla,\\gamma)=\\exp\\left(\\int_{\\Sigma_t}-F_\\nabla\\right)=\\exp\\left(\\int_{\\Sigma^c_t} F_\\nabla \\right)<br \/>\n$$<br \/>\n\u6545, \u4ee4$t\\to0$\u5f97\u5230<br \/>\n$$<br \/>\n\\exp\\left(\\int_\\Sigma F_\\nabla\\right)=1\\implies\\int_\\Sigma F_\\nabla=-2\\pi i c(L),\\quad c(L)\\in\\mathbb{Z}.<br \/>\n$$<br \/>\n<div class='latex_defn'><span class='latex_defn_h'>\u5b9a\u4e49 14<\/span> (<span class='latex_defn_name'>\u9648\u6570<\/span>)<span class='latex_defn_h'>.<\/span> \u4e0a\u8ff0\u5b9a\u4e49\u7684$c(L)$\u79f0\u4e3a\u7ebf\u4e1b\u7684<span class=\"latex_em\">\u9648\u6570<\/span>.<br \/>\n<\/div><br \/>\n<div class='latex_examp'><span class='latex_examp_h'>\u4f8b\u5b50 6<\/span> (<span class='latex_examp_name'>\u7403\u9762\u7684\u9648\u6570<\/span>)<span class='latex_examp_h'>.<\/span> \u56de\u5fc6, \u6211\u4eec\u5df2\u7ecf\u8ba1\u7b97\u5f97\u5230$F=-i dv_g$. \u6545<br \/>\n$$<br \/>\nc(TS^2)=\\frac{i}{2\\pi}\\int_{S^2}-i dv_g=2.<br \/>\n$$<br \/>\n<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u76ee\u5f55 \u76ee\u5f55&#x00A0;1.&#x00A0;&#x00A0;\u7ebf\u4e1b\u7684\u5b9a\u4e49 1.&#x00A0;\u7ebf\u4e1b\u7684\u5b9a\u4e49 \u7ebf\u4e1b&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=467\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u7ebf\u4e1b\u4e0a\u7684\u8054\u7edc\u3001\u66f2\u7387\u3001\u548c\u4e50\u7fa4\u4ee5\u53ca\u9648\u7c7b<\/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":[107,100,105,106,108],"class_list":["post-467","post","type-post","status-publish","format-standard","hentry","category-math","tag-helequn","tag-qushuai","tag-xiancong","tag-lianluo","tag-chenlei","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/467","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=467"}],"version-history":[{"count":49,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/467\/revisions"}],"predecessor-version":[{"id":530,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/467\/revisions\/530"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=467"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=467"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}