{"id":573,"date":"2018-01-09T08:24:19","date_gmt":"2018-01-09T08:24:19","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=573"},"modified":"2018-01-09T14:08:01","modified_gmt":"2018-01-09T14:08:01","slug":"fajubuyaozhengbiaojiadecunzaixing","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=573","title":{"rendered":"\u6cd5\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6\u7684\u5b58\u5728\u6027"},"content":{"rendered":"<p>\u5047\u8bbe$M^m$\u662f\u4e00\u4e2a$m$\u7ef4\u9ece\u66fc\u6d41\u5f62, $p\\in M$ \u662f\u4e00\u4e2a\u70b9\u3002\u6211\u4eec\u79f0\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6$\\{e_i\\}_{i=1}^m$\u662f$p$\u5904\u7684<span class=\"latex_em\">\u6cd5\u6b63\u4ea4\u6807\u67b6<\/span>, \u5982\u679c\u5728$p$\u5904, \u5bf9\u4efb\u610f\u7684$i,j=1,2,\\ldots,m$, \u6210\u7acb<br \/>\n$$<br \/>\n\\nabla_{e_i}e_j=0.<br \/>\n$$<br \/>\n<div class='latex_prop'><span class='latex_prop_h'>\u547d\u9898 1<\/span><span class='latex_prop_h'>.<\/span> \u5bf9\u4efb\u548c\u9ece\u66fc\u6d41\u5f62$M$\u4e0a\u7684\u4efb\u610f\u56fa\u5b9a\u70b9$p$, \u5b58\u5728$p$\u5904\u7684\u6cd5\u5e7a\u6b63\u6807\u67b6\u3002<br \/>\n<\/div><!--more--><\/p>\n<div class='latex_proof'><span class='latex_proof_h'>\u8bc1\u660e<\/span><span class='latex_proof_h'>.<\/span> \u4e8b\u5b9e\u4e0a, \u5047\u8bbe$\\{e_i\\}_{i=1}^m$\u662f$p$\u5904\u7684\u4e00\u4e2a\u5e7a\u6b63\u6807\u67b6, $x=(x^1,\\ldots,x^m)$\u662f$p$\u5904\u7531\u5e7a\u6b63\u6807\u67b6$\\{e_i\\}$\u4ee5\u53ca\u6307\u6570\u6620\u7167$\\exp_p$\u5b9a\u4e49\u7684$p$\u5904\u67d0\u90bb\u57df$U$\u4e0a\u7684\u5750\u6807\u7cfb, \u5373\u5bf9\u4efb\u610f\u7684$q\\in U\\subset M$, \u6709$q=\\exp_p(x^ie_i)$.<br \/>\n\u9996\u5148, \u6211\u4eec\u8bf4\u660e\u4e0a\u8ff0\u5750\u6807$x$\u662f$p$\u5904\u7684\u6cd5\u5750\u6807, \u5373<br \/>\n<ul><li>$g_{ij}=\\langle X_i,X_j\\rangle$\u6ee1\u8db3$g_{ij}(p)=\\delta_{ij}$;<\/li><li>$\\Gamma_{ij}^k(p)=0$; \u7279\u522b\u5730$X_k g_{ij}=0$;<\/li><\/ul>\u8bb0$X_k=\\frac{\\partial}{\\partial x^k}$\u4e3a\u81ea\u7136\u6807\u67b6, $\\gamma_i(t)=\\exp_p(te_i)$\u4e3a\u5750\u6807\u66f2\u7ebf\u3002\u6ce8\u610f\u5230, \u5207\u6620\u7167$d\\exp$\u6ee1\u8db3$d\\exp_p(e_i)=\\left.\\frac{d}{dt}\\right|_{t=0}\\gamma_i(t)=X_i(\\gamma_i(0))\\left.\\frac{d}{dt}\\right|_{t=0}t=e_i$, \u5373$d\\exp_p=\\mathrm{id}$, \u4ece\u800c<br \/>\n$$<br \/>\ng_{ij}(p)=\\langle X_i,X_j\\rangle|_p=\\langle d\\exp_{p}e_i,d\\exp_pe_j\\rangle=\\langle e_i,e_j\\rangle=\\delta_{ij}.<br \/>\n$$<\/p>\n<p>\u53c8\u6ce8\u610f\u5230\u5bf9\u4efb\u610f$p$\u5904\u7684\u5207\u5411\u91cf$v=v^ie_i$, \u4ece$p$\u51fa\u53d1\u4ee5$v$\u4e3a\u521d\u59cb\u5411\u91cf\u7684\u6d4b\u5730\u7ebf\u53ef\u8868\u793a\u4e3a$\\gamma_v(t)=\\exp_p(tv)$, \u6545\u7531\u6d4b\u5730\u7ebf\u65b9\u7a0b<br \/>\n$$<br \/>\n0=\\ddot\\gamma^k_v(t)+\\Gamma_{ij}^k(\\gamma_v(t))\\dot\\gamma_v^i(t)\\dot\\gamma_v^j(t),\\quad\\forall k=1,\\ldots,m,<br \/>\n$$<br \/>\n\u8fd9\u91cc, $\\gamma_v^k(t)=v^k t$\u662f$\\gamma_v(t)$\u7684\u5750\u6807\u5206\u91cf\u3002\u4e8e\u662f\u5728$p$\u5904<br \/>\n$$<br \/>\n0=\\Gamma_{ij}^k(p)v^iv^j.<br \/>\n$$<br \/>\n\u7531$v$\u7684\u4efb\u610f\u6027\u77e5$\\Gamma_{ij}^k(p)=0$, \u5bf9\u4efb\u610f\u7684$i,j,k=1,2\\ldots,m$\u6210\u7acb\u3002\u7531Gauss\u8f6e\u6362\u77e5$X_k g_{ij}$\u53ef\u7528$\\Gamma_{ij}^k$\u8868\u793a\u51fa\u6765, \u6545$X_k g_{ij}=0$.<\/p>\n<p>\u73b0\u5728, \u5047\u8bbe$\\{E_i\\}$\u662f$\\{e_i\\}$\u901a\u8fc7\u6cbf\u6d4b\u5730\u7ebf\u5e73\u884c\u79fb\u52a8\u5728$U$\u4e0a\u5b9a\u4e49\u7684\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6(\u5e73\u884c\u79fb\u52a8\u4fdd\u6301\u957f\u5ea6\u4e0e\u5939\u89d2)\u3002\u6211\u4eec\u5c06\u8bf4\u660e$\\{E_i\\}$\u5c31\u662f$p$\u5904\u7684\u6cd5\u5e7a\u6b63\u6807\u67b6\u3002\u4e8b\u5b9e\u4e0a, \u5047\u8bbe$E_i=f_{ik}X_k$, \u5219\u6309\u7167\u6cbf$\\gamma_l(t)$\u5e73\u884c\u79fb\u52a8\u7684\u5b9a\u4e49(\u6211\u4eec\u8bb0$\\frac{D}{dt}$\u4e3a\u6cbf\u66f2\u7ebf$\\gamma_l(t)$\u7684\u5171\u53d8\u5fae\u5206),<br \/>\n\\begin{align*}<br \/>\n0&#038;=\\frac{D}{dt}E_i=\\frac{D}{dt}(f_{ik}X_k)\\\\<br \/>\n&#038;:=\\nabla_{\\dot\\gamma_l(t)}(f_{ik}X_k)|_{\\gamma_l(t)}=[\\nabla_{X_l}(f_{ik}X_k)]|_{\\gamma_l(t)}\\\\<br \/>\n&#038;=[X_l(f_{ik})X_k+f_{ik}\\Gamma_{lk}^jX_j]|_{\\gamma_l(t)}<br \/>\n\\end{align*}<br \/>\n\u7279\u522b\u5730, \u5728$p$\u5904, \u7531\u4e8e$\\Gamma_{ij}^k(p)=\\Gamma_{ij}^k(\\gamma_l(0))=0$,<br \/>\n$$<br \/>\n[X_l(f_{ik})X_k]_{p}=0.<br \/>\n$$<br \/>\n\u8fd9\u6837, \u6211\u4eec\u76f4\u63a5\u8ba1\u7b97\u5f97\u5230, \u5728$p$\u5904<br \/>\n\\begin{align*}<br \/>\n\\nabla_{E_i}E_j&#038;=\\nabla_{f_{ik}X_k}(f_{jl}X_l)=f_{ik}(\\nabla_{X_k}(f_{jl}X_l))\\\\<br \/>\n&#038;=f_{ik}(X_k(f_{jl})X_l+f_{jl}\\nabla_{X_k}X_l)\\\\<br \/>\n&#038;=f_{ik}f_{jl}\\Gamma_{kl}^iX_i\\\\<br \/>\n&#038;=0.<br \/>\n\\end{align*}<br \/>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5047\u8bbe$M^m$\u662f\u4e00\u4e2a$m$\u7ef4\u9ece\u66fc\u6d41\u5f62, $p\\in M$ \u662f\u4e00\u4e2a\u70b9\u3002\u6211\u4eec\u79f0\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6$\\{e_i\\}_{i=&hellip; <a class=\"more-link\" href=\"https:\/\/blog.vanabel.cn\/?p=573\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6cd5\u5c40\u90e8\u5e7a\u6b63\u6807\u67b6\u7684\u5b58\u5728\u6027<\/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":[119,121,120,101],"class_list":["post-573","post","type-post","status-publish","format-standard","hentry","category-math","tag-yaozhengbiaojia","tag-zhishuyingzhao","tag-fazuobiao","tag-huodongbiaojia","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/573","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=573"}],"version-history":[{"count":3,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/573\/revisions"}],"predecessor-version":[{"id":576,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/573\/revisions\/576"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=573"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=573"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}