{"id":3740,"date":"2013-09-05T10:32:46","date_gmt":"2013-09-05T02:32:46","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=3740"},"modified":"2013-09-05T10:32:46","modified_gmt":"2013-09-05T02:32:46","slug":"%e6%b4%bb%e5%8a%a8%e6%a0%87%e6%9e%b6%e7%9a%84%e7%bb%93%e6%9e%84%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=3740","title":{"rendered":"\u6d3b\u52a8\u6807\u67b6\u7684\u7ed3\u6784\u65b9\u7a0b"},"content":{"rendered":"

\u5047\u8bbe$M$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62, $\\set{e_i}$\u662f\u4e00\u4e2a\u5c40\u90e8\u6807\u67b6\u573a, \u5176\u5bf9\u5076\u6807\u67b6\u573a\u8bb0\u4e3a$\\set{\\omega^i}$. \u5219\u6211\u4eec\u6709\u5982\u4e0b\u7684\u7ed3\u6784\u65b9\u7a0b:
\n\\[\\begin{cases}
\n\\rd \\omega^i=-\\omega^i_j\\wedge\\omega^j+\\frac{1}{2}T_{kl}^i\\omega^k\\wedge\\omega^l\\\\
\n\\rd \\omega^i_j=-\\omega^i_k\\wedge\\omega^k_j+\\Omega_j^i,
\n\\end{cases}\\]
\n\u5176\u4e2d$\\omega^i_j$\u662f\u8054\u7edc1\u5f62\u5f0f<\/strong>(\u56de\u5fc6, \u7ed9\u5b9a\u4e00\u4e2a\u8054\u7edc$\\nabla$, \u4ee5\u53ca\u4e00\u4e2a\u5c40\u90e8\u6807\u67b6\u573a$\\set{e_i}$, \u8054\u7edc1\u5f62\u5f0f$\\set{\\omega^i_j}$\u7531\u4e0b\u5f0f\u5b9a\u4e49:$\\nabla_X(e_j)=\\omega^i_j(X)e_i$.), $T_{kl}^i$\u662f\u6320\u7387\u5f20\u91cf<\/strong>\u7684\u5206\u91cf(\u56de\u5fc6, \u5047\u8bbe$M$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62, $\\nabla$\u662f\u5176\u4e0a\u4efb\u4e00\u8054\u7edc<\/strong>. \u5b9a\u4e49\u6320\u7387\u5f20\u91cf$T$\u5982\u4e0b: $T(X,Y)=\\nabla_XY-\\nabla_YX-[X,Y]$, $X,Y\\in\\Gamma(TM)$, \u8fd9\u91cc$[X,Y]$\u662f$X,Y$\u7684\u674e\u62ec\u53f7, \u5b83\u628a\u4e24\u4e2a\u5149\u6ed1\u5411\u91cf\u573a\u6620\u5c04\u4e3a\u4e00\u4e2a\u5149\u6ed1\u5411\u91cf\u573a, \u5176\u5b9a\u4e49\u4e3a$[X,Y]f=XYf-YXf$, \u8fd9\u91cc$f$\u4e3a$M$\u4e0a\u7684\u5149\u6ed1\u51fd\u6570), \u5373$T(e_k,e_l)=T_{kl}^i e_i=\\frac{1}{2}T_{pq}^i\\omega^p\\wedge\\omega^q(e_k,e_l)e_i$, \u8fd9\u91cc\u89c4\u5b9a$T_{pq}^i=-T^i_{qp}$.
\n
\n$\\Omega_j^i$\u79f0\u4e3a\u66f2\u7387\u5f62\u5f0f. \u56de\u5fc6, $(1,3)$\u578b\u66f2\u7387\u5f20\u91cf\u5b9a\u4e49\u4e3a
\n\\[
\nR(X,Y)Z=\\nabla_X\\nabla_YZ-\\nabla_Y\\nabla_XZ-\\nabla_{[X,Y]}Z,
\n\\]
\n\u5176\u5206\u91cf\u7531$R(e_i,e_j)e_k=R_{ijk}^le_l$\u5f97\u5230, \u5373$R_{ijk}^l=\\omega^l(R(e_i,e_j)e_k)$. \u7c7b\u4f3c$T_{kl}^i=\\frac{1}{2}T_{pq}^i\\omega^p\\wedge\\omega^q(e_k,e_l)e_i$. \u6211\u4eec\u4e5f\u6709, $R_{klj}^i=\\frac{1}{2}R_{pqj}^i\\omega^p\\wedge\\omega^q(e_k,e_l)$, \u8fd9\u91cc\u7684$\\Omega_j^i=\\frac{1}{2}R_{klj}^i\\omega^k\\wedge\\omega^l$.<\/p>\n

\u800c$(0,4)$\u66f2\u7387\u5f20\u91cf$R$\u5b9a\u4e49\u4e3a
\n\\[
\nR(X,Y,Z,W)=g(R(X,Y)Z,W),
\n\\]
\n\u5176\u5206\u91cf\u4e3a$R_{ijkl}=R(e_i,e_j,e_k,e_l)=g(R(e_i,e_j)e_k,e_l)=g(R_{ijk}^pe_p,e_l)=g_{pl}R^p_{ijk}$.<\/p>\n

\u9996\u5148\u6765\u770b\u7b2c\u4e00\u4e2a\u5f0f\u5b50, \u6211\u4eec\u53ea\u9700\u5bf9\u5c06\u5176\u4f5c\u7528\u5230\u57fa\u4e0a\u5373\u53ef:
\n\\begin{align*}
\n\\rd\\omega^i(e_k,e_l)&=e_k\\omega^i(e_l)-e_l\\omega^i(e_k)-\\omega^i([e_k,e_l])\\\\
\n&=\\nabla_{e_k}(\\omega^i(e_l))-\\nabla_{e_l}(\\omega^i(e_k))-\\omega^i([e_k,e_l])\\\\
\n&=(\\nabla_{e_k}\\omega^i)(e_l)+\\omega^i(\\nabla_{e_k}e_l)-(\\nabla_{e_l}\\omega^i)(e_k)\\\\
\n&\\qquad-\\omega^i(\\nabla_{e_l}e_k)-\\omega^i([e_k,e_l])\\\\
\n&=-\\omega^i_j(e_k)\\omega^j(e_l)+\\omega^i_j(e_l)\\omega^j(e_k)+\\omega^i(\\nabla_{e_k}e_l)\\\\
\n&\\qquad-\\omega^i(\\nabla_{e_l}e_k)-\\omega^i([e_k,e_l])\\\\
\n&=-\\omega^i_j\\wedge\\omega^j(e_k,e_l)+\\omega^i(T(e_k,e_l))\\\\
\n&=\\left(-\\omega^i_j\\wedge\\omega^j+\\frac{1}{2}T_{pq}^i\\omega^p\\wedge\\omega^q\\right)(e_k,e_l).
\n\\end{align*}
\n\u5176\u6b21, \u6765\u770b\u7b2c\u4e8c\u5f0f, \u540c\u6837\u5c06\u5176\u4f5c\u7528\u5230\u57fa\u4e0a:
\n\\begin{align*}
\n\\rd\\omega^i_j(e_k,e_l)&=e_k(\\omega^i_j(e_l))-e_l(\\omega^i_j(e_k))-\\omega^i_j([e_k,e_l])\\\\
\n&=\\nabla_{e_k}(\\omega^i(\\nabla_{e_l}e_j))-\\nabla_{e_l}(\\omega^i(\\nabla_{e_k}e_j))-\\omega^i_j([e_k,e_l])\\\\
\n&=(\\nabla_{e_k}\\omega^i)(\\nabla_{e_l}e_j)+\\omega^i(\\nabla_{e_k}(\\nabla_{e_l}e_j))\\\\
\n&\\qquad-
\n(\\nabla_{e_l}\\omega^i)(\\nabla_{e_k}e_j)-\\omega^i(\\nabla_{e_l}(\\nabla_{e_k}e_j))-\\omega^i(\\nabla_{[e_k,e_l]}e_j)\\\\
\n&=-\\omega^i_p(e_k)\\omega^p(\\omega^q_j(e_l)e_q)+\\omega^i_p(e_l)\\omega^p(\\omega_j^q(e_k)e_q)\\\\
\n&\\qquad+\\omega^i(\\nabla_{e_k}\\nabla_{e_l}e_j-\\nabla_{e_l}\\nabla_{e_k}e_j-\\nabla_{[e_k,e_l]}e_j)\\\\
\n&=-\\omega^i_p(e_k)\\omega_j^p(e_l)+\\omega^i_p(e_l)\\omega^p_j(e_k)+\\omega^i(R(e_k,e_l)e_j)\\\\
\n&=(-\\omega^i_p\\wedge\\omega^p_j)(e_k,e_l)+\\omega^i(R(e_k,e_l)e_j)\\\\
\n&=(-\\omega^i_p\\wedge\\omega^p_j)(e_k,e_l)+R^i_{klj}\\\\
\n&=(-\\omega^i_p\\wedge\\omega^p_j+\\Omega_j^i)(e_k,e_l).
\n\\end{align*}<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5047\u8bbe$M$\u662f\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62, $\\set{e_i}$\u662f\u4e00\u4e2a\u5c40\u90e8\u6807\u67b6\u573a, \u5176\u5bf9\u5076\u6807\u67b6 …<\/p>\n

\u6d3b\u52a8\u6807\u67b6\u7684\u7ed3\u6784\u65b9\u7a0b<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[747,315],"class_list":["post-3740","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-747","tag-315"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/3740","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\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3740"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/3740\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3740"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3740"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3740"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}