{"id":4211,"date":"2011-04-22T00:22:16","date_gmt":"2011-04-21T16:22:16","guid":{"rendered":"https:\/\/lttt.blog.ustc.edu.cn\/?p=4211"},"modified":"2011-04-22T00:22:16","modified_gmt":"2011-04-21T16:22:16","slug":"%e5%87%a0%e4%bd%95%e5%8c%96%e7%8c%9c%e6%b5%8bgeometrization-conjecture","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=4211","title":{"rendered":"\u51e0\u4f55\u5316\u731c\u6d4b(Geometrization Conjecture)"},"content":{"rendered":"

Thurston\u7684\u51e0\u4f55\u5316\u731c\u6d4b\u8bf4\u4efb\u4f55\u4e00\u4e2a\u7d27\u81f4\u7684\u4e09\u7ef4\u6d41\u5f62\u90fd\u53ef\u4ee5\u81ea\u7136\u7684\u5206\u89e3\u4e3a\u542b\u6709\u51e0\u4f55\u7ed3\u6784\u7684\u5b50\u6d41\u5f62. \u5b83\u662f\u5173\u4e8e\u66f2\u9762\u7684\u4e00\u81f4\u5316\u5b9a\u7406(uniformization theorem)\u5728\u4e09\u7ef4\u60c5\u5f62\u7684\u7c7b\u4f3c\u63a8\u5e7f. \u5b83\u9996\u5148\u7531William Thurston \u4e8e1982\u5e74\u63d0\u51fa, \u800c\u540e\u53c8\u5f15\u53d1\u4e86\u8bf8\u591a\u731c\u6d4b, \u6bd4\u5982\u5e9e\u52a0\u83b1\u731c\u6d4b(Poincare Conjecture) \u548c Thurston \u7684\u692d\u5706\u5316\u731c\u6d4b(elliptization conjecture).<\/span><\/p>\n

Thurston\u7684\u53cc\u66f2\u5316\u5b9a\u7406(hyperbolization theorem)\u8868\u660eHaken\u6d41\u5f62\u6ee1\u8db3\u51e0\u4f55\u5316\u731c\u6d4b. Thurston\u5728\u4ed61980\u5e74\u7684\u8bba\u6587\u4e2d\u7ed9\u51fa\u4e86\u4e00\u4e2a\u8bc1\u660e, \u800c\u540e\u53c8\u6709\u5f88\u591a\u8865\u5145\u7684\u8bc1\u660e.<\/p>\n

Grigori Perelman\u57282003\u5e74\u7528Ricci\u6d41+\u62d3\u6251\u624b\u672f\u7684\u65b9\u6cd5\u7ed9\u51fa\u4e86\u51e0\u4f55\u5316\u731c\u6d4b\u7684\u8bc1\u660e\u7684\u4e00\u4e2a\u6897\u6982.\u800c\u4eca, \u5173\u4e8e\u51e0\u4f55\u5316\u731c\u6d4b\u7684\u8bc1\u660e\u6709\u56db\u672c\u4e0d\u540c\u7684\u8bba\u8457. \u5e9e\u52a0\u83b1\u731c\u6d4b\u4ee5\u53ca\u7403\u7a7a\u95f4\u5f62\u5f0f(Spherical space form conjecture)\u731c\u6d4b\u90fd\u662f\u51e0\u4f55\u5316\u731c\u6d4b\u7684\u63a8\u8bba, \u5c3d\u7ba1\u5173\u4e8e\u5e9e\u52a0\u83b1\u731c\u6d4b\u7684\u4e00\u4e9b\u7b80\u77ed\u7684\u8bc1\u660e\u5e76\u4e0d\u80fd\u7528\u4e8e\u51e0\u4f55\u5316\u731c\u6d4b.<\/p>\n

\u5177\u4f53\u800c\u8a00, \u51e0\u4f55\u5316\u731c\u6d4b\u662f\u8fd9\u6837\u7684:
\n\u9996\u5148\u770b\u4e24\u4e2a\u6982\u5ff5<\/p>\n

    \n
  1. \u4e00\u4e2a\u4e09\u7ef4\u6d41\u5f62\u79f0\u4e3a\u662f\u95ed\u7684\u5982\u679c\u5b83\u662f\u7d27\u81f4\u65e0\u8fb9\u7684\u6d41\u5f62.<\/li>\n
  2. \u6bcf\u4e00\u4e2a\u95ed\u7684\u4e09\u7ef4\u6d41\u5f62\u90fd\u6709\u4e00\u4e2a\u672c\u539f\u5206\u89e3(prime decomposition), \u5373\u5b83\u90fd\u662f\u4e00\u4e9b\u672c\u539f\u4e09\u7ef4\u6d41\u5f62\u7684\u8fde\u901a\u548c(\u8fd9\u79cd\u5206\u89e3\u5f53\u7136\u662f\u552f\u4e00\u7684\u9664\u4e86\u5728\u975e\u5b9a\u5411\u6d41\u5f62\u65f6\u7684\u4e00\u70b9\u5c0f\u95ee\u9898). \u8fd9\u5c06\u4e09\u7ef4\u6d41\u5f62\u7684\u7814\u7a76\u5f52\u7ed3\u4e3a\u672c\u539f\u4e09\u7ef4\u6d41\u5f62\u7684\u7814\u7a76: \u5373\u53ea\u9700\u7814\u7a76\u90a3\u4e9b\u4e0d\u80fd\u5199\u6210\u5e73\u51e1\u8fde\u901a\u548c\u7684\u6d41\u5f62.<\/li>\n<\/ol>\n

    \u73b0\u5728\u6765\u53d9\u8ff0Thurston\u7684\u51e0\u4f55\u5316\u731c\u6d4b:
    \n\u6bcf\u4e00\u4e2a\u5b9a\u5411\u7684\u4e09\u7ef4\u95ed\u6d41\u5f62\u90fd\u53ef\u4ee5\u6cbf\u7740\u73af\u5207\u5f00, \u4f7f\u5f97\u5207\u5f00\u540e\u7684\u6bcf\u4e2a\u5b50\u6d41\u5f62\u7684\u5185\u90e8\u90fd\u6709\u4e00\u4e2a\u5177\u6709\u6709\u9650\u4f53\u79ef\u7684\u51e0\u4f55\u7ed3\u6784(\u9ece\u66fc\u5ea6\u91cf).<\/p>\n

    \u6240\u6709\u7684\u4e09\u7ef4\u6d41\u5f62\u53ef\u80fd\u67098\u79cd\u51e0\u4f55\u4f53. \u8fd9\u4e9b\u6a21\u578b\u51e0\u4f55\u4f53\u662f\u4e00\u4e2a\u5149\u6ed1\u3001\u8fde\u901a\u7684\u6d41\u5f62X, \u800c\u4e14\u5728\u5176\u4e0a\u8fd8\u6709\u4e00\u4e2a\u5177\u6709\u7d27\u7a33\u5b9a\u5b50\u7684\u674e\u7fa4G\u53ef\u8fc1\u7684\u4f5c\u7528.<\/p>\n

    \u4e00\u4e2a\u6a21\u578b\u51e0\u4f55\u4f53\u79f0\u4e3a\u662f\u6781\u5927\u7684, \u5982\u679cG\u662fX\u4e0a\u6240\u6709\u5177\u6709\u7d27\u7a33\u5b9a\u5b50\u7684\u5149\u6ed1\u53ef\u8fc1\u4f5c\u7528\u4e2d\u6781\u5927\u7684. \u6709\u65f6\u5728\u6a21\u578b\u51e0\u4f55\u4f53\u7684\u5b9a\u4e49\u4e2d\u4e5f\u8981\u6c42\u8fd9\u4e00\u6761.<\/p>\n

    \u6d41\u5f62M\u4e0a\u7684\u4e00\u4e2a\u51e0\u4f55\u7ed3\u6784\u662f\u4eceM\u5230X\/T\u7684\u4e00\u4e2a\u5fae\u5206\u540c\u80da. \u5176\u4e2dX\u662f\u4e00\u4e2a\u6a21\u578b\u51e0\u4f55\u4f53, \u800cT\u662fG\u7684\u4e00\u4e2a\u79bb\u6563\u5b50\u7fa4, \u5b83\u81ea\u7531\u7684\u4f5c\u7528\u5728X\u4e0a. \u5982\u679c\u7ed9\u5b9a\u7684\u6d41\u5f62\u4e0a\u5bb9\u8bb8\u4e00\u4e2a\u51e0\u4f55\u7ed3\u6784, \u90a3\u4e48\u5c06\u6709\u4e00\u4e2a\u4f7f\u5b83\u7684\u6a21\u578b\u51e0\u4f55\u4f53\u662f\u6781\u5927\u7684.<\/p>\n

    \u4e00\u4e2a\u4e09\u7ef4\u7684\u6a21\u51e0\u4f55\u4f53X\u79f0\u4e3a\u4e0e\u51e0\u4f55\u5316\u731c\u6d4b\u662f\u76f8\u5173\u7684, \u5982\u679c\u5b83\u662f\u6781\u5927\u7684, \u800c\u4e14\u81f3\u5c11\u5b58\u5728\u4e00\u4e2a\u7d27\u81f4\u6d41\u5f62, \u5176\u51e0\u4f55\u7ed3\u6784\u4f4d\u4e8e\u6a21\u578b\u51e0\u4f55\u4f53\u662fX\u4e0a. Thurston\u5c06\u6ee1\u8db3\u6761\u4ef6\u7684\u6a21\u51e0\u4f55\u4f53\u5206\u4e3a8\u7c7b, \u5b83\u4eec\u6709\u65f6\u4e5f\u79f0\u4e3aThurston\u51e0\u4f55\u4f53.(\u6ce8\u610f\u5982\u679c\u4e0d\u8981\u6c42\u5176\u5546\u662f\u7d27\u81f4\u7684, \u90a3\u4e48\u5c06\u6709\u4e0d\u53ef\u6570\u591a\u7684\u6a21\u51e0\u4f55\u4f53)<\/p>\n

    \u6a21\u51e0\u4f55\u4f53\u4e0eBianchi\u7fa4\u4e5f\u6709\u4e00\u4e9b\u8054\u7cfb, Bianchi\u7fa4\u6307\u4e09\u7ef4\u674e\u7fa4. \u5927\u591a\u6570\u7684Thurston\u51e0\u4f55\u4f53\u90fd\u53ef\u4ee5\u901a\u8fc7Bianchi\u7fa4\u4e0a\u4e00\u4e2a\u5de6\u4e0d\u53d8\u7684\u5ea6\u91cf\u6765\u5b9e\u73b0. \u7136\u800c S^2\u00d7R\u5374\u4e0d\u80fd, \u6b27\u6c0f\u7a7a\u95f4\u6709\u4e24\u4e2a\u4e0d\u540c\u7684
    \nBianchi\u7fa4, \u800c\u4e14\u5b58\u5728\u4e0d\u53ef\u6570\u591a\u4e2a\u975e\u5e7a\u6a21\u53ef\u89e3Bianchi\u7fa4, \u5b83\u4eec\u4e2d\u7684\u5927\u591a\u6570\u7ed9\u51fa\u5177\u6709\u975e\u7d27\u6a21\u51e0\u4f55\u4f53\u7684\u8868\u793a.
    \n8\u79cdThurston\u6a21\u51e0\u4f55\u4f53\u662f:<\/p>\n

      \n
    1. \u6807\u51c6\u7403\u9762S^3\uff0c\u5177\u6709\u5e38\u66f2\u7387+l<\/li>\n
    2. \u6b27\u6c0f\u7a7a\u95f4E^3\uff0c\u5177\u6709\u5e38\u66f2\u73870<\/li>\n
    3. \u53cc\u66f2\u7a7a\u95f4H^3\uff0c\u5177\u6709\u5e38\u66f2\u7387-1<\/li>\n
    4. S^2\u00d7R<\/li>\n
    5. H^2\u00d7R<\/li>\n
    6. \u7279\u6b8a\u7ebf\u6027\u7fa4Sl_2(R)\u4e0a\u7684\u6574\u4f53\u8986\u76d6<\/li>\n
    7. \u5e42\u96f6\u51e0\u4f55<\/li>\n
    8. \u53ef\u89e3\u51e0\u4f55<\/li>\n<\/ol>\n

      \u672c\u539f\u5b9a\u5411\u4e09\u7ef4\u6d41\u5f62\u53ef\u6309\u552f\u4e00\u7684\u4e00\u79cd\u6781\u5c0f\u5316\u65b9\u6cd5\u6cbf\u7740\u73af\u5207\u5272\u6210\u7247\u72b6\u6d41\u5f62, \u5373Seifert\u6d41\u5f62\u6216\u8005atoroidal, \u8fd9\u79f0\u4e3aJSJ\u5206\u89e3(JSJ decomposition), \u800c\u5728\u51e0\u4f55\u5316\u731c\u6d4b\u4e2d\u76f8\u5e94\u7684\u5206\u89e3\u4e0d\u662f\u552f\u4e00\u7684, \u56e0\u4e3a\u5728JSJ\u5206\u89e3\u4e2d\u7684\u7247\u72b6\u6d41\u5f62\u53ef\u80fd\u4e0d\u5177\u6709\u6709\u9650\u4f53\u79ef.(\u4f8b\u5982, \u4e00\u4e2a\u73af\u7684Anosov\u6620\u5c04\u4e0b\u7684\u50cf(\u4ecd\u662f\u4e00\u4e2a\u73af)\u5177\u6709\u6709\u9650\u7684\u53ef\u89e3\u7ed3\u6784(sol structure), \u4f46\u662f\u5b83\u7684JSJ\u5206\u89e3\u5c06\u5176\u6cbf\u7740\u4e00\u4e2a\u73af\u5256\u5f00\u6210\u4e00\u4e2a\u73af\u4e0e\u5355\u4f4d\u533a\u95f4\u7684\u4e58\u79ef,\u800c\u8fd9\u4e2a\u4e58\u79ef\u7684\u5185\u90e8\u6ca1\u6709\u6709\u9650\u4f53\u79ef\u7684\u7ed3\u6784.)<\/p>\n

      \u5bf9\u4e8e\u975e\u5b9a\u5411\u6d41\u5f62\u7684\u51e0\u4f55\u5316\u731c\u6d4b, \u6700\u7b80\u5355\u7684\u63d0\u6cd5\u662f\u8003\u8651\u5b83\u7684\u6709\u54112\u91cd\u8986\u76d6. \u5f53\u7136, \u4e5f\u53ef\u4ee5\u76f4\u63a5\u8003\u8651\u975e\u5b9a\u5411\u6d41\u5f62, \u4f46\u662f\u8fd9\u4f1a\u989d\u5916\u7684\u589e\u52a0\u590d\u6742\u5ea6:\u53ef\u80fd\u9700\u8981\u6cbf\u7740\u5c04\u5f71\u5e73\u9762(projective plane),\u514b\u83b1\u56e0\u74f6(Klein bottles), \u7403(spheres)\u4ee5\u53ca\u73af\u9762(tori)\u6765\u5256\u5206. \u800c\u4e14\u5177\u6709\u5c04\u5f71\u5e73\u9762\u4f5c\u4e3a\u8fb9\u754c\u7684\u6d41\u5f62\u901a\u5e38\u6ca1\u6709\u51e0\u4f55\u7ed3\u6784, \u8fd9\u5c31\u989d\u5916\u5730\u589e\u52a0\u4e86\u4e00\u4e9b\u590d\u6742\u5ea6.<\/p>\n

      \u58f0\u660e:\u672c\u6587\u8bd1\u81f3wiki.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Thurston\u7684\u51e0\u4f55\u5316\u731c\u6d4b\u8bf4\u4efb\u4f55\u4e00\u4e2a\u7d27\u81f4\u7684\u4e09\u7ef4\u6d41\u5f62\u90fd\u53ef\u4ee5\u81ea\u7136\u7684\u5206\u89e3\u4e3a\u542b\u6709\u51e0\u4f55\u7ed3 …<\/p>\n

      \u51e0\u4f55\u5316\u731c\u6d4b(Geometrization Conjecture)<\/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":[],"class_list":["post-4211","post","type-post","status-publish","format-standard","hentry","category-mathnotes"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4211","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=4211"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4211\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}