{"id":866,"date":"2021-07-22T01:21:28","date_gmt":"2021-07-22T01:21:28","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=866"},"modified":"2021-07-22T01:21:51","modified_gmt":"2021-07-22T01:21:51","slug":"limanjihedeyixietimu","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=866","title":{"rendered":"\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u9898\u76ee"},"content":{"rendered":"

\u8fd9\u4e9b\u9898\u76ee\u6765\u81ea\u5929\u5143\u897f\u5357\u6570\u5b66\u4e2d\u5fc3\u7684\u9752\u5e74\u6559\u5e08\u6691\u671f\u57f9\u8bad.<\/p>\n

  1. \u5047\u8bbe$(M,g)$\u662f$n\\geq3$\u7ef4\u8fde\u901a\u9ece\u66fc\u6d41\u5f62, \u82e5\u5b58\u5728$\\lambda\\in C^\\infty(M)$\u4f7f\u5f97$\\mathrm{Ric}_M=\\lambda g$, \u8bc1\u660e:
    \n
    1. $M$\u662f\u4e00\u4e2aEinstein\u6d41\u5f62, \u5373$M$\u7684\u6570\u91cf\u66f2\u7387\u662f\u5e38\u6570;
      \n <\/li>
    2. \u5f53$n=3$\u65f6, $M$\u662f\u5e38\u66f2\u7387\u7a7a\u95f4;
      \n <\/li>
    3. \u82e5$M$\u7684\u6570\u91cf\u66f2\u7387\u4e0d\u4e3a\u96f6, \u5219$M$\u4e0a\u4e0d\u5b58\u5728\u975e\u96f6\u7684\u5e73\u884c\u5411\u91cf\u573a.
      \n <\/li><\/ol>
    4. \u5047\u8bbe$R_+^n=\\left\\{ x=(x^1,\\ldots,x^n)\\in \\mathbb{R}^n:x^n>0 \\right\\}$. \u8003\u5bdf\u5176\u4e0a\u7684\u9ece\u66fc\u5ea6\u91cf$g$,
      \n \\[
      \n g_{ij}=
      \n \\begin{cases}
      \n 1\/(x^n)^2,&i=j=1,2,\\ldots,n,\\\\
      \n 0,&i\\neq j.
      \n \\end{cases}
      \n \\]
      \n
      1. \u6c42$(\\mathbb{R}_{+}^n,g)$\u7684\u622a\u9762\u66f2\u7387;
        \n <\/li>
      2. \u6c42$(\\mathbb{R}_+^n,g)$\u4e0a\u8fc7\u70b9$(0,0,\\ldots,0,1)$\u4e14\u521d\u59cb\u5207\u5411\u91cf\u4e3a\u5355\u4f4d\u5411\u91cf$\\nu$(\u5916\u6cd5\u5411)\u7684\u6d4b\u5730\u7ebf;
        \n <\/li>
      3. \u8bc1\u660e$(\\mathbb{R}_+^n,g)$\u662f\u5b8c\u5907\u7684;
        \n <\/li>
      4. \u8bc1\u660e$(\\mathbb{R}_+^n,g)$\u5728\u4efb\u4f55\u4e00\u70b9\u7684\u6307\u6570\u6620\u5c04\u662f\u5fae\u5206\u540c\u80da\u7684.
        \n <\/li><\/ol>
      5. \u5047\u8bbe$M$\u662f\u5177\u6709\u6b63\u622a\u9762\u66f2\u7387\u7684$n$\u7ef4\u7d27\u9ece\u66fc\u6d41\u5f62. \u8bc1\u660e:
        \n
        1. $n$\u4e3a\u5076\u6570\u65f6, \u5982\u679c$M$\u53ef\u5b9a\u5411, \u5219\u5b83\u662f\u5355\u8fde\u901a\u7684; \u5982\u679c$M$\u4e0d\u53ef\u5b9a\u5411, \u5219$\\pi_1(M)\\cong \\mathbb{Z}_2$;
          \n <\/li>
        2. $n$\u4e3a\u57fa\u6570\u65f6, \u5219$M$\u53ef\u5b9a\u5411. \u5e76\u4e3e\u4f8b\u8bf4\u660e$M$\u4e0d\u4e00\u5b9a\u662f\u5355\u8fde\u901a\u7684.
          \n <\/li><\/ol>
        3. \u5047\u8bbe$(M,g)$\u662f\u8fde\u901a\u7684\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, $M$\u4e2d\u4ece$x\\in M$\u51fa\u53d1, \u4ee5\u5f27\u957f\u4e3a\u53c2\u6570\u7684\u6d4b\u5730\u7ebf$\\gamma:[0,+\\infty)\\to M$\u79f0\u4e3a\u4ece$x$\u51fa\u53d1\u7684\u5c04\u7ebf, \u5982\u679c$\\gamma(0)=x$, \u4e14\u5bf9\u4efb\u610f\u7684$t\\in[0,+\\infty)$, $\\gamma|_{[0,t]}$\u662f\u8fde\u63a5$x$\u4e0e$\\gamma(t)$\u7684\u6700\u77ed\u66f2\u7ebf, \u5373$d(x,\\gamma(t))=t$. \u8bc1\u660e: $M$\u662f\u975e\u7d27\u7684\u5145\u8981\u6761\u4ef6\u662f\u5bf9\u4efb\u610f\u7684$x\\in M$, \u5728$M$\u4e0a\u90fd\u6709\u4ece$x$\u51fa\u53d1\u7684\u5c04\u7ebf.
          \n <\/li>
        4. \u5047\u8bbe$M$\u662f\u53ef\u5b9a\u5411\u7684\u95ed\u6d41\u5f62, $f\\in C^\\infty(M)$\u662f\u6b21\u8c03\u548c\u51fd\u6570, \u5373$\\Delta_Mf\\leq0$, \u8bc1\u660e$f$\u4e00\u5b9a\u662f\u5e38\u503c\u51fd\u6570.
          \n <\/li>
        5. \u5047\u8bbe$M$\u662f\u5355\u8fde\u901a\u5b8c\u5907\u9ece\u66fc\u6d41\u5f62, \u5bf9\u4efb\u610f\u7684$p\\in M$, \u82e5$p$\u70b9\u6cbf\u6240\u6709\u4ece$p$\u51fa\u53d1\u7684\u5f84\u5411\u6d4b\u5730\u7ebf\u7684\u7b2c\u4e00\u5171\u8f6d\u70b9\u90fd\u662f\u540c\u4e00\u70b9$q$, \u4e14$p\\neq q$, $d(p,q)=\\pi$. \u8bc1\u660e: \u5982\u679c$K_M\\leq 1$, \u5219$M$\u4e0e\u6807\u51c6\u7403$S^n$\u7b49\u8ddd.<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

          \u8fd9\u4e9b\u9898\u76ee\u6765\u81ea\u5929\u5143\u897f\u5357\u6570\u5b66\u4e2d\u5fc3\u7684\u9752\u5e74\u6559\u5e08\u6691\u671f\u57f9\u8bad. \u5047\u8bbe$(M,g)$\u662f$n\\geq3$\u7ef4\u8fde\u901a\u9ece\u66fc\u6d41\u5f62, \u82e5\u5b58\u5728… \u7ee7\u7eed\u9605\u8bfb\u9ece\u66fc\u51e0\u4f55\u7684\u4e00\u4e9b\u9898\u76ee<\/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":[126],"class_list":["post-866","post","type-post","status-publish","format-standard","hentry","category-math","tag-limanjihe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/866","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=866"}],"version-history":[{"count":1,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/866\/revisions"}],"predecessor-version":[{"id":867,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/866\/revisions\/867"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=866"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=866"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=866"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}