{"id":319,"date":"2013-06-28T00:31:58","date_gmt":"2013-06-27T16:31:58","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=319"},"modified":"2013-06-28T00:31:58","modified_gmt":"2013-06-27T16:31:58","slug":"%e8%bd%ac%e8%bd%bdimpa%e9%bb%8e%e6%9b%bc%e5%87%a0%e4%bd%95%e8%80%83%e8%af%95%e9%a2%98","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=319","title":{"rendered":"[\u8f6c\u8f7d]IMPA\u9ece\u66fc\u51e0\u4f55\u8003\u8bd5\u9898"},"content":{"rendered":"<h4 id=\"1\">\u671f\u4e2d\u8003\u8bd5<a id=\"fnref:mid\" class=\"footnote\" title=\"See footnote\" href=\"#fn:mid\">1<\/a><\/h4>\n<ol>\n<li>\n<ul>\n<li>\u5047\u8bbe$X,Y\\in \\R^n$, $\\nabla$ \u662f\u6807\u51c6\u6b27\u6c0f\u5ea6\u91cf\u4e0b\u7684Levi-Civita\u8054\u7edc, \u8bd5\u6c42$\\nabla_XY$ (\u7528\u5176\u5206\u91cf\u8868\u793a), \u5e76\u8bc1\u660e$\\nabla$\u662f$\\R^n$\u4e2d\u552f\u4e00\u4e00\u4e2a\u5e73\u884c\u7684\u7ebf\u6027\u8054\u7edc.<\/li>\n<li>\u5047\u8bbe$(M^n,g)$\u548c$(N^n,h)$\u662f\u4e24\u4e2a\u9ece\u66fc\u6d41\u5f62, \u5176\u66f2\u7387\u5f20\u91cf\u5206\u522b\u8bb0\u4e3a$R_M$, $R_N$. \u8bc1\u660e\u5bf9\u4efb\u4e00\u7b49\u8ddd$f:M\\to N$, \u90fd\u6709<br \/>\n\\[<br \/>\n\\langle R_N(\\rd f(X),\\rd f(Y))\\rd f(Z),\\rd f(W)\\rangle_h=\\langle R_M(X,Y)Z,W\\rangle_g.<br \/>\n\\]<!--more--><\/li>\n<\/ul>\n<\/li>\n<li>\n<ul>\n<li>\u8bc1\u660e\u4efb\u4e00\u5fae\u5206\u6d41\u5f62\u4e0a\u90fd\u5b58\u5728\u4e00\u4e2a\u5b8c\u5907\u7684\u9ece\u66fc\u5ea6\u91cf.<\/li>\n<li>\u4e3e\u4f8b\u8bf4\u660e\u5b58\u5728\u4e00\u4e2a\u975e\u7d27\u5b8c\u5907\u7684\u9ece\u66fc\u6d41\u5f62, \u5176\u6d4b\u5ea6\u662f\u6709\u9650\u7684.<\/li>\n<\/ul>\n<\/li>\n<li>\n<ul>\n<li>\u8bc1\u660e\u9ad8\u65af\u516c\u5f0f.<\/li>\n<li>\u5047\u8bbe$(M^n,g)$\u662f\u4e00\u5177\u6709\u8d1f\u622a\u9762\u66f2\u7387\u7684\u9ece\u66fc\u6d41\u5f62, \u8bc1\u660e, \u5f53$n\\geq3$\u65f6\u5b83\u4e0d\u53ef\u7b49\u8ddd\u5d4c\u5165\u5230\u6b27\u6c0f\u7a7a\u95f4$\\R^{n+1}$\u4e2d.<\/li>\n<\/ul>\n<\/li>\n<li>\u5047\u8bbe$(M,g)$\u662f\u4e00\u5b8c\u5907\u5355\u8fde\u901a\u7684\u9ece\u66fc\u6d41\u5f62, \u4e14\u5177\u6709\u975e\u6b63\u622a\u9762\u66f2\u7387. \u8bd5\u5bf9\u6d4b\u5730\u4e09\u89d2\u5f62\u8bc1\u660e\u5982\u4e0b\u7684\u4f59\u5f26\u4e0d\u7b49\u5f0f<br \/>\n\\[<br \/>\na^2\\geq b^2+c^2-2bc\\cos\\angle\\langle b,c\\rangle.<br \/>\n\\]<\/li>\n<\/ol>\n<h4 id=\"2\">\u671f\u672b\u8003\u8bd5<a id=\"fnref:final\" class=\"footnote\" title=\"See footnote\" href=\"#fn:final\">2<\/a><\/h4>\n<ol>\n<li>\u5047\u8bbe$\\gamma:[0,a]\\to M$\u4e3a\u4e00\u6700\u77ed\u6d4b\u5730\u7ebf, \u4e14$\\gamma(a)$\u4e0e$\\gamma(b)$\u4e0d\u5171\u8f6d. \u4ee4$V$\u662f\u6cbf\u7740$\\gamma$\u7684\u4e00\u4e2a\u5206\u6bb5\u5149\u6ed1\u5411\u91cf\u573a, \u800c$I$\u4e3a\u6307\u6807\u5f62\u5f0f.\n<ul>\n<li>\u8bc1\u660e\u5b58\u5728\u6cbf$\\gamma$\u7684Jacobi\u573a$J$, \u4f7f\u5f97$J(0)=V(0)$, $J(a)=V(a)$.<\/li>\n<li>\u8bc1\u660e$I(J,J)\\leq I(V,V)$, \u4e14\u7b49\u53f7\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53$J=V$.<\/li>\n<li>\u5047\u8bbe$\\beta$\u662f\u53e6\u4e00\u4f7f\u5f97$\\beta(0)$\u4e0e$\\beta(a)$\u5171\u8f6d\u7684\u6d4b\u5730\u7ebf. \u8bc1\u660e: \u5bf9\u4efb\u610f\u7684$\\eps&gt;0$, $\\beta:[0,a+\\eps]\\to M$\u8fd9\u6bb5\u6d4b\u5730\u7ebf\u7684\u6307\u6570\u5fc5\u4e3a\u6b63.<\/li>\n<\/ul>\n<\/li>\n<li>\u5047\u8bbe$(M,g)$\u4e3a\u4e00\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, $r(x)$\u4e3a$x$\u5230\u57fa\u70b9$p$\u7684\u8ddd\u79bb.\n<ul>\n<li>\u5982\u679c$x\\in(M\\setminus\\set{p})\\bigcup\\mathrm{Cut}(p)$, \u90a3\u4e48$r(x)$\u5728$x$\u7684\u4e00\u4e2a\u90bb\u57df\u4e0a\u53ef\u5fae, \u4e14$\\|\\nabla r(x)\\|=1$.<\/li>\n<li>\u5047\u8bbe$\\gamma:[0,l]\\to M$\u662f\u4e00\u8fde\u63a5$p$\u4e0e$x\\in(M\\setminus\\set{p})\\bigcup\\mathrm{Cut}(p)$\u7684\u6b63\u89c4\u6d4b\u5730\u7ebf. \u5bf9\u4efb\u610f\u7684$v\\in T_xM$, $v\\perp\\gamma'(l)$, \u8bb0$J_v$\u4e3a\u6cbf\u7740$\\gamma$\u7684Jacobi\u573a, \u4e14$J_v(0)=0$, $J_v(l)=v$. \u8bd5\u8bc1\u660e:<br \/>\n\\[<br \/>\n(\\mathrm{Hess})_x(v,v)=I(J_v,J_v).<br \/>\n\\]<\/li>\n<li>\u5047\u8bbe$M$\u7684\u622a\u9762\u66f2\u7387$K\\leq\\delta$, $\\delta\\in\\R$. $\\gamma$\u5982\u4e0a\u95ee\u4e2d\u5b9a\u4e49, \u4e14\u6ee1\u8db3: \u5f53$\\delta&gt;0$\u65f6, $l&lt;\\frac{\\pi}{\\sqrt\\delta}$. \u8bd5\u6bd4\u8f83$M$\u4e0e\u622a\u9762\u66f2\u7387\u4e3a$\\delta$\u7684\u7a7a\u95f4\u5f62\u5f0f$\\tilde M$\u4e0a\u8ddd\u79bb\u51fd\u6570\u7684Hessian\u4e4b\u95f4\u7684\u5173\u7cfb.<\/li>\n<\/ul>\n<\/li>\n<li>\n<ul>\n<li>\u5047\u8bbe$(M.g)$\u4e3a\u4e00\u7d27\u81f4\u9ece\u66fc\u6d41\u5f62, \u4e14\u5176\u622a\u66f2\u7387$K&lt;0$. \u8bc1\u660e: $M$\u4e0a\u4efb\u610f\u4e24\u6761\u81ea\u7531\u540c\u4f26\u7684\u95ed\u6d4b\u5730\u7ebf\u5fc5\u76f8\u540c.<\/li>\n<li>\u5047\u8bbe$M,N$\u662f\u4e24\u4e2a\u7d27\u81f4\u7684\u5fae\u5206\u6d41\u5f62, \u8bc1\u660e$M\\times N$\u4e0a\u4e0d\u5b58\u5728\u622a\u66f2\u7387$K&lt;0$\u7684\u9ece\u66fc\u5ea6\u91cf.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h4 id=\"\">\u53c2\u8003\u6559\u6750<\/h4>\n<ol>\n<li>Do Carmo, Manfredo P. <em>Riemannian geometry<\/em>. Birkh\u00e4user Boston, 1992.<\/li>\n<li>Petersen, Peter. <em>Riemannian geometry<\/em>. Vol. 171. New York: Springer, 2006.<\/li>\n<\/ol>\n<blockquote><p>Written with <a href=\"http:\/\/benweet.github.io\/stackedit\/\">StackEdit<\/a>.<\/p><\/blockquote>\n<h4>Footnotes<\/h4>\n<div class=\"footnotes\">\n<hr \/>\n<ol>\n<li id=\"fn:mid\"><a href=\"http:\/\/www.liuxiaochuan.org\/2012\/05\/midterm.htm\">http:\/\/www.liuxiaochuan.org\/2012\/05\/midterm.htm<\/a> <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:mid\">\u21a9<\/a><\/li>\n<li id=\"fn:final\"><a href=\"http:\/\/www.liuxiaochuan.org\/2012\/07\/riemanniangeometry.htm\">http:\/\/www.liuxiaochuan.org\/2012\/07\/riemanniangeometry.htm<\/a> <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:final\">\u21a9<\/a><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u671f\u4e2d\u8003\u8bd51 \u5047\u8bbe$X,Y\\in \\R^n$, $\\nabla$ \u662f\u6807\u51c6\u6b27\u6c0f\u5ea6\u91cf\u4e0b &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=319\"> <span class=\"screen-reader-text\">[\u8f6c\u8f7d]IMPA\u9ece\u66fc\u51e0\u4f55\u8003\u8bd5\u9898<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/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":[513,418,428],"class_list":["post-319","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-impa","tag-418","tag-428"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/319","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=319"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/319\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}