{"id":116899,"date":"2026-10-22T08:24:43","date_gmt":"2026-10-22T00:24:43","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/116899.html"},"modified":"2026-10-22T08:24:43","modified_gmt":"2026-10-22T00:24:43","slug":"%e6%b7%b7%e6%b2%8c%e6%8b%93%e6%89%91%e5%ad%a6hdt%e4%b8%89%e7%bb%b4%e5%94%af%e4%b8%80%e5%8e%9f%e7%90%86-%e5%af%b9%e5%bc%a6%e8%ae%ba%e9%ab%98%e7%bb%b4%e5%81%87%e8%af%b4%e7%9a%84%e5%89%96%e6%9e%90","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/116899.html","title":{"rendered":"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790"},"content":{"rendered":"<h3>\u6458\u8981<\/h3>\n<p>\u672c\u6587\u5728\u6df7\u6c8c\u62d3\u6251\u5b66\u7684\u6846\u67b6\u4e0b&#xff0c;\u63d0\u51fa\u5e76\u4e25\u683c\u8bc1\u660e\u4e00\u6761\u57fa\u672c\u5b9a\u7406&#xff1a;\u4e09\u7ef4\u662f\u552f\u4e00\u7ef4\u5ea6\u3002\u7ef4\u5ea6\u552f\u4e00\u6027<\/p>\n<p>\u8bba\u8bc1\u7684\u6838\u5fc3&#xff0c;\u662f\u628a&#034;\u7ef4\u5ea6&#034;\u7684\u5224\u636e\u4ece&#034;\u65b9\u5411\u7684\u6761\u6570&#034;\u6539\u4e3a&#034;\u80fd\u5426\u4e0e\u65e2\u6709\u65b9\u5411\u6b63\u4ea4&#034;\u3002\u5728\u6b64\u5224\u636e\u4e0b&#xff1a;<\/p>\n<ul>\n<li>\u4e00\u7ef4&#xff1a;\u6b63\u4ea4\u662f\u4e8c\u5143\u5173\u7cfb&#xff0c;\u4e00\u7ef4\u4e2d\u65e0\u4ece\u5efa\u7acb&#xff0c;\u6545\u9000\u5316&#xff1b;<\/li>\n<li>\u4e8c\u7ef4&#xff1a;\u7f3a\u7b2c\u4e09\u6761\u4e92\u76f8\u6b63\u4ea4\u7684\u65b9\u5411&#xff0c;\u65e0\u6cd5\u754c\u5b9a&#034;\u9762\u5185&#xff0f;\u9762\u5916&#034;&#xff0c;\u6545\u4e0d\u53ef\u81ea\u8db3&#xff1b;<\/li>\n<li>\u4e09\u7ef4&#xff1a;\u6b63\u4ea4\u57fa\u5b8c\u6574&#xff0c;\u4e14\u7531\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406\u77e5\u5df2\u7a77\u5c3d&#xff0c;\u6545\u81ea\u6d3d\u95ed\u5408&#xff1b;<\/li>\n<li>\u9ad8\u7ef4&#xff08;n \u2265 4&#xff09;&#xff1a;\u82e5\u5185\u5207\u4e8e\u4e09\u7ef4&#xff0c;\u5219\u5fc5\u4e0e\u67d0\u4e00\u57fa\u65b9\u5411\u975e\u6b63\u4ea4&#xff0c;\u56e0\u800c\u9000\u5316\u4e3a\u65e2\u6709\u57fa\u7684\u7ebf\u6027\u7ec4\u5408&#xff0c;\u4e0d\u643a\u5e26\u65b0\u7684\u72ec\u7acb\u4fe1\u606f&#xff1b;\u82e5\u4e0d\u81ea\u5207\u4e8e\u4e09\u7ef4&#xff0c;\u5219\u4e0d\u53ef\u6784\u9020\u3002<\/li>\n<\/ul>\n<p>\u672c\u6587\u7279\u522b\u8bba\u8bc1&#xff1a;\u6b63\u4ea4\u5224\u636e\u7684\u4e25\u683c\u5f62\u5f0f\u662f\u5185\u79ef\u4e3a\u96f6&#xff0c;\u5b83\u4e0e\u89d2\u5ea6\u5355\u4f4d\u7684\u9009\u53d6\u65e0\u5173\u3002\u6240\u8c13&#034;90\u00b0&#034;\u4ec5\u662f\u8fd9\u4e00\u4e0d\u53d8\u91cf\u5728 360 \u5ea6\u5236\u4e0b\u7684\u8bfb\u6570\u6807\u7b7e&#xff0c;\u4e0d\u6784\u6210\u5224\u636e\u672c\u8eab\u3002<\/p>\n<p>\u672c\u6587\u8fdb\u4e00\u6b65\u7ed9\u51fa\u56db\u6761\u72ec\u7acb\u7684\u62d3\u6251\u5b66\u8bc1\u636e\u4e0e\u4e09\u6761\u7269\u7406\u5b66\u8bc1\u636e\u2014\u2014\u7ebd\u7ed3\u7684\u552f\u4e00\u627f\u8f7d\u6027\u3001Whitney \u624b\u672f\u7684\u7ef4\u6570\u95e8\u69db\u3001Poincar\u00e9 \u96be\u5ea6\u8c31\u3001\u53c9\u79ef\u95ed\u5408\u6027&#xff0c;\u4ee5\u53ca Bertrand \u5b9a\u7406\u5bf9\u7a33\u5b9a\u8f68\u9053\u7684\u552f\u4e00\u6311\u9009\u2014\u2014\u5b83\u4eec\u4e00\u81f4\u5730\u6307\u5411\u540c\u4e00\u4e2a\u7ed3\u8bba&#xff1a;\u4e09\u7ef4\u4e0d\u662f\u4f17\u591a\u7ef4\u5ea6\u4e2d\u7684\u4e00\u4e2a&#xff0c;\u800c\u662f\u7ef4\u5ea6\u6982\u5ff5\u5f97\u4ee5\u6210\u7acb\u7684\u552f\u4e00\u821e\u53f0\u3002<\/p>\n<p>\u5173\u952e\u8bcd&#xff1a;\u6df7\u6c8c\u62d3\u6251\u5b66&#xff1b;\u4e09\u7ef4\u552f\u4e00\u539f\u7406&#xff1b;\u6b63\u4ea4\u5224\u636e&#xff1b;\u5ea6\u5236\u65e0\u5173\u6027&#xff1b;\u6b63\u4ea4\u5bb9\u91cf&#xff1b;\u5185\u5207\u9000\u5316&#xff1b;\u7ebd\u7ed3&#xff1b;\u4e34\u754c\u7ef4\u5ea6<\/p>\n<hr \/>\n<h3>1 \u5f15\u8a00<\/h3>\n<h4>1.1 \u95ee\u9898\u7684\u63d0\u51fa<\/h4>\n<p>\u5728\u73b0\u4ee3\u6570\u5b66\u7684\u901a\u884c\u7528\u6cd5\u4e2d&#xff0c;\u7ef4\u5ea6\u88ab\u89c6\u4e3a\u4e00\u4e2a\u53ef\u81ea\u7531\u53d6\u503c\u7684\u53c2\u6570&#xff1a;R\u00b9\u3001R\u00b2\u3001R\u00b3\u3001R\u2074\u2026\u2026\u76f4\u81f3\u65e0\u7a77\u7ef4\u7684 Hilbert \u7a7a\u95f4&#xff0c;\u90fd\u88ab\u8ba4\u4e3a\u662f\u540c\u7b49\u826f\u5b9a\u4e49\u7684\u6570\u5b66\u5bf9\u8c61\u3002\u6309 Brouwer \u7ef4\u6570\u4e0d\u53d8\u6027\u5b9a\u7406&#xff08;1911&#xff09;&#xff0c;R^m \u4e0e R^n \u5728 m \u2260 n \u65f6\u4e0d\u540c\u80da&#xff0c;\u56e0\u6b64\u6570\u5b66\u4e0a\u7684\u786e\u5b58\u5728\u4e00\u4e2a\u7531\u65e0\u7a77\u591a\u4e2a\u4e92\u4e0d\u76f8\u540c\u7a7a\u95f4\u6784\u6210\u7684&#034;\u7ef4\u5ea6\u65cf&#034;\u3002<\/p>\n<p>\u7136\u800c&#xff0c;\u4e00\u65e6\u628a\u7ef4\u5ea6\u7406\u89e3\u4e3a\u7269\u7406\u4e16\u754c\u7684\u7ed3\u6784\u5c5e\u6027\u800c\u975e\u63cf\u8ff0\u5de5\u5177\u7684\u53c2\u6570&#xff0c;\u4e0a\u8ff0\u56fe\u666f\u4fbf\u7acb\u523b\u51fa\u73b0\u95ee\u9898&#xff1a;\u6211\u4eec\u4ece\u672a\u89c2\u6d4b\u5230\u3001\u4e5f\u4ece\u672a\u6784\u9020\u51fa\u4e00\u4e2a\u771f\u5b9e\u7684\u7b2c\u56db\u7a7a\u95f4\u65b9\u5411&#xff1b;\u6211\u4eec\u4e5f\u4ece\u672a\u8fdb\u5165\u8fc7\u4e00\u4e2a\u539a\u5ea6\u4e25\u683c\u4e3a\u96f6\u7684\u4e8c\u7ef4\u4e16\u754c\u3002<\/p>\n<p>\u4e8e\u662f\u4ea7\u751f\u5982\u4e0b\u95ee\u9898&#xff1a;<\/p>\n<p>\u5728\u4f55\u79cd\u4e25\u683c\u610f\u4e49\u4e0b&#xff0c;\u4e09\u7ef4\u662f\u552f\u4e00\u7684\u7ef4\u5ea6&#xff1f;<\/p>\n<h4>1.2 \u6df7\u6c8c\u62d3\u6251\u5b66\u7684\u4f4d\u7f6e<\/h4>\n<p>\u6df7\u6c8c\u62d3\u6251\u5b66\u4ee5\u6df7\u6c8c\u672c\u4f53\u4e3a\u7814\u7a76\u5bf9\u8c61\u3002\u8fd9\u91cc\u9700\u8981\u5148\u6f84\u6e05\u4e00\u4e2a\u5173\u952e\u8bbe\u5b9a&#xff1a;\u6df7\u6c8c\u4e0d\u662f\u65e0\u5e8f&#xff0c;\u800c\u662f\u4e0d\u53ef\u89e3\u5f00\u7684\u76f8\u4e92\u5173\u8054\u3002\u6240\u8c13&#034;\u4e0d\u53ef\u89e3\u5f00&#034;&#xff0c;\u6307\u4e0d\u5b58\u5728\u4efb\u4f55\u8fde\u7eed\u53d8\u5f62\u80fd\u628a\u8fd9\u79cd\u5173\u8054\u62c6\u6563\u2014\u2014\u8fd9\u6b63\u662f\u62d3\u6251\u5b66\u6240\u7814\u7a76\u7684&#034;\u7f20\u7ed5\u6027\u8d28&#034;\u3002\u6df7\u6c8c\u4f5c\u4e3a\u672c\u4f53&#xff0c;\u5176\u7ed3\u6784\u7684\u6839\u672c\u7279\u5f81\u6b63\u662f\u8fd9\u79cd\u7f20\u7ed5\u4e0e\u7ea0\u7f20\u3002<br \/>\n\u62d3\u6251\u5b66\u6b63\u662f\u7814\u7a76&#034;\u5728\u8fde\u7eed\u53d8\u5f62\u4e0b\u4fdd\u6301\u4e0d\u53d8\u7684\u7f20\u7ed5\u6027\u8d28&#034;\u7684\u5b66\u79d1\u3002\u4e8e\u662f\u95ee\u9898\u8f6c\u5316\u4e3a&#xff1a;\u4ec0\u4e48\u6837\u7684\u7a7a\u95f4\u7ef4\u5ea6&#xff0c;\u80fd\u627f\u8f7d\u8fd9\u79cd\u4e0d\u53ef\u89e3\u5f00\u7684\u7f20\u7ed5&#xff1f; \u672c\u6587\u7684\u4e00\u4e2a\u5173\u952e\u53d1\u73b0\u662f&#xff1a;\u7f20\u7ed5\u672c\u8eab\u662f\u4e09\u7ef4\u72ec\u6709\u7684\u73b0\u8c61\u3002<\/p>\n<ul>\n<li>\u5728 n \u2264 2 \u7ef4\u4e2d&#xff0c;\u95ed\u5408\u66f2\u7ebf\u65e0\u6cd5\u5f62\u6210\u7ebd\u7ed3&#xff1b;<\/li>\n<li>\u5728 n &#061; 3 \u7ef4\u4e2d&#xff0c;\u5b58\u5728\u65e0\u7a77\u591a\u4e2a\u4e92\u4e0d\u7b49\u4ef7\u7684\u3001\u4e0d\u53ef\u89e3\u5f00\u7684\u7ebd\u7ed3&#xff1b;<\/li>\n<li>\u5728 n \u2265 4 \u7ef4\u4e2d&#xff0c;\u4e00\u5207\u7ebd\u7ed3\u5747\u53ef\u88ab\u89e3\u5f00&#xff0c;\u7ea0\u7f20\u81ea\u52a8\u6d88\u5931\u3002<\/li>\n<\/ul>\n<p>\u6df7\u6c8c\u82e5\u8981\u4f5c\u4e3a\u771f\u5b9e\u7ed3\u6784\u800c\u5b58\u5728&#xff0c;\u5b83\u5fc5\u987b\u80fd\u627f\u8f7d\u4e0d\u53ef\u89e3\u5f00\u7684\u7ea0\u7f20&#xff1b;\u800c\u552f\u4e00\u80fd\u627f\u8f7d\u7ea0\u7f20\u7684\u7ef4\u5ea6\u662f\u4e09\u7ef4\u3002<\/p>\n<h4>1.3 \u8bba\u8bc1\u7b56\u7565<\/h4>\n<li>\u7b2c 2 \u8282&#xff1a;\u628a\u7ef4\u5ea6\u7684\u5224\u636e\u4e25\u683c\u5316\u4e3a\u6b63\u4ea4\u6027&#xff0c;\u5e76\u8bba\u8bc1\u8be5\u5224\u636e\u4e0e\u89d2\u5ea6\u5355\u4f4d\u65e0\u5173&#xff1b;<\/li>\n<li>\u7b2c 3 \u8282&#xff1a;\u8bc1\u660e\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406\u4e0e\u65b9\u5411\u4f59\u5f26\u5b88\u6052\u5f8b&#xff0c;\u5e76\u7ed9\u51fa\u6b63\u4ea4\u4e8f\u635f\u7684\u53ef\u91cf\u5316\u4e0b\u754c&#xff1b;<\/li>\n<li>\u7b2c 4 \u8282&#xff1a;\u8bc1\u660e\u4e09\u7ef4\u4e2d\u65b9\u5411\u6709\u65e0\u7a77\u591a&#xff0c;\u4f46\u6b63\u4ea4\u57fa\u81f3\u591a\u4e09\u4e2a&#xff0c;\u4ece\u800c\u5bfc\u51fa\u5185\u5207\u9000\u5316\u5b9a\u7406&#xff1b;<\/li>\n<li>\u7b2c 5 \u8282&#xff1a;\u8bc1\u660e\u4e00\u7ef4\u3001\u4e8c\u7ef4\u7684\u9000\u5316\u6027&#xff1b;<\/li>\n<li>\u7b2c 6 \u8282&#xff1a;\u4ece\u7ebd\u7ed3\u3001Whitney \u624b\u672f\u3001Poincar\u00e9 \u96be\u5ea6\u8c31\u3001\u53c9\u79ef\u95ed\u5408\u6027\u56db\u4e2a\u65b9\u5411\u7ed9\u51fa\u62d3\u6251\u5b66\u8bc1\u636e&#xff1b;<\/li>\n<li>\u7b2c 7 \u8282&#xff1a;\u7ed9\u51fa\u7269\u7406\u5b66\u8bc1\u636e&#xff1b;<\/li>\n<li>\u7b2c 8 \u8282&#xff1a;\u9648\u8ff0\u4e09\u7ef4\u552f\u4e00\u539f\u7406\u53ca\u5176\u63a8\u8bba\u3002<\/li>\n<hr \/>\n<p>\u4e0b\u8868\u5c06\u7b2c 2 \u81f3\u7b2c 8 \u8282\u7684\u8bba\u8bc1\u73af\u8282\u3001\u6838\u5fc3\u5b9a\u7406\/\u8bc1\u636e\u4e0e\u7ed3\u8bba\u6c47\u603b\u4e3a\u4e00\u5f20\u300c\u8bba\u8bc1\u8def\u7ebf\u56fe\u300d&#xff0c;\u4fbf\u4e8e\u8bfb\u8005\u628a\u63e1\u4ece\u6b63\u4ea4\u5224\u636e\u5230\u4e09\u7ef4\u552f\u4e00\u539f\u7406\u7684\u5b8c\u6574\u903b\u8f91\u94fe\u6761&#xff1a;<\/p>\n<table>\n<tr>\u8bba\u8bc1\u73af\u8282\u6838\u5fc3\u5b9a\u7406 \/ \u8bc1\u636e\u7ed3\u8bba<\/tr>\n<tbody>\n<tr>\n<td>\u00a72 \u5224\u636e\u786e\u7acb<\/td>\n<td>\u5b9a\u4e49 2.1&#xff08;\u7ef4\u5ea6&#061;\u4e24\u4e24\u6b63\u4ea4\u65b9\u5411\u7684\u6700\u5927\u6570\u76ee&#xff09;&#xff1b;\u5224\u636e 2.2&#xff08;\u6b63\u4ea4\u5224\u636e&#xff09;&#xff1b;\u547d\u9898 2.3&#xff08;\u5ea6\u5236\u65e0\u5173\u6027&#xff09;&#xff1b;\u6781\u5316\u6052\u7b49\u5f0f&#xff08;\u5185\u79ef\u4e3a\u96f6&#xff09;<\/td>\n<td>\u7ef4\u5ea6\u7531\u5185\u79ef\u4e3a\u96f6\u8fd9\u4e00\u4e0d\u53d8\u91cf\u523b\u753b&#xff0c;\u4e0e\u89d2\u5ea6\u5355\u4f4d\u65e0\u5173<\/td>\n<\/tr>\n<tr>\n<td>\u00a73 \u5bb9\u91cf\u4e0a\u754c<\/td>\n<td>\u5f15\u7406 3.1&#xff08;\u6b63\u4ea4\u7ec4\u7ebf\u6027\u65e0\u5173&#xff09;&#xff1b;\u5b9a\u7406 3.2&#xff08;\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406&#xff09;&#xff1b;\u5b9a\u7406 3.5&#xff08;\u65b9\u5411\u4f59\u5f26\u5b88\u6052\u5f8b&#xff09;&#xff1b;\u5b9a\u7406 3.7&#xff08;\u6b63\u4ea4\u4e8f\u635f\u4e0b\u754c&#xff09;<\/td>\n<td>R\u00b3 \u4e2d\u4e24\u4e24\u6b63\u4ea4\u65b9\u5411\u81f3\u591a 3 \u4e2a&#xff1b;\u4efb\u4f55\u65b9\u5411\u81f3\u5c11\u4e0e\u67d0\u57fa\u65b9\u5411\u5185\u79ef \u2265 1\/\u221a3<\/td>\n<\/tr>\n<tr>\n<td>\u00a74 \u65b9\u5411\u65e0\u7a77\u4e0e\u5185\u5207\u9000\u5316<\/td>\n<td>\u5b9a\u7406 4.1&#xff08;\u65b9\u5411\u8fde\u7eed\u7edf&#xff09;&#xff1b;\u5b9a\u7406 4.2&#xff08;\u5185\u5207\u9000\u5316\u5b9a\u7406&#xff09;&#xff1b;\u63a8\u8bba 4.4&#xff08;\u9ad8\u7ef4\u4e0d\u53ef\u6784\u9020\u6027&#xff09;<\/td>\n<td>\u4e09\u7ef4\u542b\u65e0\u7a77\u591a\u65b9\u5411\u4f46\u6b63\u4ea4\u5bb9\u91cf\u4ec5 3&#xff1b;\u989d\u5916\u65b9\u5411\u5185\u5207\u5373\u9000\u5316\u4e3a\u7ebf\u6027\u7ec4\u5408&#xff0c;\u4e0d\u6784\u6210\u65b0\u7ef4\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>\u00a75 \u4f4e\u7ef4\u9000\u5316<\/td>\n<td>\u547d\u9898 5.1&#xff08;\u4e00\u7ef4\u6b63\u4ea4\u65e0\u4ece\u5efa\u7acb&#xff09;&#xff1b;\u547d\u9898 5.3&#xff08;\u4e8c\u7ef4\u5185\u5916\u4e0d\u53ef\u754c\u5b9a&#xff09;&#xff1b;\u5b9a\u7406 5.5&#xff08;\u4f4e\u7ef4\u9000\u5316\u5b9a\u7406&#xff09;<\/td>\n<td>\u4e00\u7ef4\u3001\u4e8c\u7ef4\u5206\u522b\u662f R\u00b3 \u7684\u4e00\u7ef4\/\u4e8c\u7ef4\u622a\u9762&#xff0c;\u4e0d\u53ef\u81ea\u8db3<\/td>\n<\/tr>\n<tr>\n<td>\u00a76 \u62d3\u6251\u5b66\u8bc1\u636e<\/td>\n<td>\u4e8b\u5b9e 6.1&#xff08;\u7ebd\u7ed3\u552f\u4e00\u627f\u8f7d\u6027&#xff09;&#xff1b;\u4e8b\u5b9e 6.2&#xff08;Whitney \u624b\u672f\u95e8\u69db&#xff09;&#xff1b;\u4e8b\u5b9e 6.3&#xff08;Poincar\u00e9 \u96be\u5ea6\u8c31&#xff09;&#xff1b;\u4e8b\u5b9e 6.4&#xff08;\u53c9\u79ef\u95ed\u5408\u6027&#xff09;&#xff1b;\u5b9a\u7406 6.5&#xff08;\u4e34\u754c\u7ef4\u5ea6\u5b9a\u7406&#xff09;<\/td>\n<td>\u4e09\u7ef4\u662f\u552f\u4e00\u80fd\u9501\u5b9a\u4e0d\u53ef\u89e3\u5f00\u7ea0\u7f20\u3001\u4e14\u65e0\u4f59\u91cf\u5177\u521a\u6027\u7684\u4e34\u754c\u7ef4\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>\u00a77 \u7269\u7406\u5b66\u8bc1\u636e<\/td>\n<td>\u9ad8\u65af\u5b9a\u5f8b&#xff08;F \u221d 1\/r^{n\u22121}&#xff09;&#xff1b;Bertrand \u5b9a\u7406&#xff08;\u552f\u4e00\u6311\u9009 n &#061; 3&#xff09;&#xff1b;Ehrenfest \u8bba\u8bc1&#xff08;\u91cf\u5b50\u7a33\u5b9a\u6027&#xff09;<\/td>\n<td>\u7269\u7406\u5224\u636e\u5728\u7ef4\u5ea6\u65cf\u4e2d\u552f\u4e00\u6311\u51fa n &#061; 3&#xff0c;\u5141\u8bb8\u7a33\u5b9a\u675f\u7f1a\u7ed3\u6784<\/td>\n<\/tr>\n<tr>\n<td>\u00a78 \u539f\u7406\u7efc\u5408<\/td>\n<td>\u4e09\u7ef4\u552f\u4e00\u539f\u7406&#xff08;\u9000\u5316\u6027\u3001\u95ed\u5408\u6027\u3001\u5305\u542b\u6027\u3001\u552f\u4e00\u6027&#xff09;&#xff1b;\u63a8\u8bba 8.1\u20138.4<\/td>\n<td>\u4f4e\u7ef4\u662f\u4e09\u7ef4\u7684\u9000\u5316\u6295\u5f71&#xff0c;\u9ad8\u7ef4\u662f\u4e09\u7ef4\u7684\u7b26\u53f7\u53e0\u52a0&#xff1b;\u4e09\u7ef4\u662f\u7ef4\u5ea6\u6982\u5ff5\u5f97\u4ee5\u6210\u7acb\u7684\u552f\u4e00\u821e\u53f0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>2 \u9884\u5907&#xff1a;\u7ef4\u5ea6\u7684\u6b63\u4ea4\u5224\u636e<\/h3>\n<h4>2.1 &#034;\u5b58\u5728&#034;\u7684\u4e09\u4e2a\u5c42\u6b21<\/h4>\n<table>\n<tr>\u5c42\u6b21\u5224\u636e\u8bf4\u660e<\/tr>\n<tbody>\n<tr>\n<td>\u6570\u5b66\u5b58\u5728<\/td>\n<td>\u65e0\u77db\u76fe\u5373\u53ef\u6784\u9020<\/td>\n<td>\u5f62\u5f0f\u4e3b\u4e49\u5224\u636e&#xff1b;R^n \u5bf9\u4e00\u5207 n \u6210\u7acb<\/td>\n<\/tr>\n<tr>\n<td>\u6784\u9020\u5b58\u5728<\/td>\n<td>\u53ef\u5728\u65e2\u5b9a\u7a7a\u95f4\u5185\u88ab\u5b9e\u9645\u6784\u9020<\/td>\n<td>\u672c\u6587\u7684\u6838\u5fc3\u5224\u636e<\/td>\n<\/tr>\n<tr>\n<td>\u7269\u7406\u5b58\u5728<\/td>\n<td>\u53ef\u5b9e\u73b0\u4e14\u5fae\u6270\u7a33\u5b9a<\/td>\n<td>\u89c1\u7b2c 7 \u8282<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u672c\u6587\u7684\u4e3b\u8981\u7ed3\u8bba\u5c5e\u4e8e\u6784\u9020\u5c42\u6b21\u3002\u8fd9\u91cc\u9700\u8981\u5f3a\u8c03\u5c42\u6b21\u5dee\u5f02&#xff1a;\u6570\u5b66\u5b58\u5728&#xff08;R\u2074\u3001R\u2075 \u7b49\u826f\u5b9a\u4e49&#xff09;\u4e0e\u6784\u9020\u5b58\u5728&#xff08;\u5728\u4e09\u7ef4\u73b0\u5b9e\u4e2d\u53ef\u88ab\u5b9e\u9645\u6784\u9020&#xff09;\u662f\u4e24\u56de\u4e8b\u3002\u672c\u6587\u5e76\u4e0d\u5426\u8ba4\u524d\u8005&#xff1b;\u672c\u6587\u8bc1\u660e\u7684\u662f\u540e\u8005&#xff1a;\u9ad8\u7ef4\u7a7a\u95f4\u65e0\u6cd5\u5728\u4e09\u7ef4\u73b0\u5b9e\u4e2d\u88ab\u6784\u9020\u51fa\u6765&#xff1b;\u4e00\u65e6\u8bd5\u56fe\u5185\u5207\u6784\u9020&#xff0c;\u5176\u6b63\u4ea4\u6027\u5fc5\u7136\u4e27\u5931\u3002<\/p>\n<h4>2.2 \u6b63\u4ea4\u5224\u636e<\/h4>\n<p>\u5b9a\u4e49 2.1&#xff08;\u7ef4\u5ea6&#xff09;\u3000\u8bbe V \u662f\u4e00\u5b9e\u5185\u79ef\u7a7a\u95f4\u3002V \u7684\u7ef4\u5ea6 dim V&#xff0c;\u5b9a\u4e49\u4e3a V \u4e2d\u4e24\u4e24\u6b63\u4ea4\u7684\u975e\u96f6\u65b9\u5411\u7684\u6700\u5927\u6570\u76ee\u3002<\/p>\n<p>\u5224\u636e 2.2&#xff08;\u6b63\u4ea4\u5224\u636e&#xff09;\u3000\u4e00\u4e2a\u65b9\u5411 u \u76f8\u5bf9\u4e8e\u65b9\u5411\u96c6\u5408 {v\u2081, \u2026, v_k} \u6784\u6210\u65b0\u7ef4\u5ea6&#xff0c;\u5f53\u4e14\u4ec5\u5f53<\/p>\n<p>u \u00b7 v\u1d62 &#061; 0\u3000\u5bf9\u6240\u6709 i &#061; 1, \u2026, k \u6210\u7acb\u3002<\/p>\n<p>\u5373 u \u4e0e\u65e2\u6709\u8bf8\u65b9\u5411\u6b63\u4ea4\u3002<\/p>\n<h4>2.3 \u4e3a\u4ec0\u4e48\u5224\u636e\u662f&#034;\u6b63\u4ea4&#034;&#xff0c;\u800c\u4e0d\u662f&#034;90\u00b0&#034;<\/h4>\n<p>\u547d\u9898 2.3&#xff08;\u5ea6\u5236\u65e0\u5173\u6027\u5b9a\u7406&#xff09;\u3000\u6b63\u4ea4\u5173\u7cfb\u4e0e\u89d2\u5ea6\u5355\u4f4d\u7684\u9009\u53d6\u65e0\u5173\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u8bbe\u5468\u5929\u88ab\u7b49\u5206\u4e3a N \u4efd&#xff0c;\u6bcf\u4efd\u79f0\u4e3a&#034;1 \u5355\u4f4d\u89d2&#034;\u3002\u8bb0 \u03c6_N(u, v) \u4e3a u\u3001v \u5939\u89d2\u5728\u8be5\u5ea6\u5236\u4e0b\u7684\u8bfb\u6570\u3002<\/p>\n<p>\u6b63\u4ea4 &#061; \u5939\u89d2\u4e3a\u5468\u5929\u7684\u56db\u5206\u4e4b\u4e00&#xff0c;\u5373&#xff1a;<\/p>\n<p>\u03c6_N &#061; N \/ 4<\/p>\n<p>\u4e8e\u662f&#xff1a;N &#061; 360 \u65f6\u8bfb\u6570\u4e3a 90&#xff1b;N &#061; 400 \u65f6\u8bfb\u6570\u4e3a 100&#xff1b;N &#061; 2\u03c0 \u65f6\u8bfb\u6570\u4e3a \u03c0\/2&#xff1b;N &#061; 1 \u65f6\u8bfb\u6570\u4e3a 1\/4\u3002<\/p>\n<p>\u8bfb\u6570\u968f N \u53d8\u5316&#xff0c;\u800c&#034;\u56db\u5206\u4e4b\u4e00\u5468\u5929&#034;\u8fd9\u4e00\u4f4d\u7f6e\u4e0d\u53d8\u3002 \u6545\u6b63\u4ea4\u5173\u7cfb\u4e0d\u4f9d\u8d56\u4e8e\u5ea6\u5236\u7684\u9009\u62e9\u3002\u220e<\/p>\n<p>\u63a8\u8bba 2.4\u3000&#034;\u5939\u89d2\u7b49\u4e8e 90\u00b0&#034;\u4e0d\u662f\u5224\u636e&#xff0c;\u800c\u662f\u6b63\u4ea4\u5173\u7cfb\u5728 360 \u5ea6\u5236\u4e0b\u7684\u4e00\u4e2a\u8bfb\u6570\u6807\u7b7e\u3002\u4ee5\u6570\u503c&#034;90&#034;\u4f5c\u4e3a\u5224\u636e&#xff0c;\u5c06\u4f7f\u5224\u636e\u4f9d\u8d56\u4e8e\u4e00\u4e2a\u4efb\u610f\u7ea6\u5b9a\u2014\u2014\u6362\u4e00\u5957\u5ea6\u5236\u5373\u5931\u6548\u3002\u8fd9\u662f\u628a\u4e0d\u53d8\u91cf\u7684\u5173\u7cfb\u8bef\u5f53\u6210\u4e86\u7ea6\u5b9a\u6027\u7684\u8bfb\u6570\u3002\u56e0\u6b64&#xff0c;\u672c\u6587\u5168\u90e8\u8bba\u8bc1\u5747\u4ee5&#034;\u5185\u79ef\u4e3a\u96f6&#034;\u8fd9\u4e00\u4e0d\u53d8\u91cf\u4e3a\u51c6&#xff0c;\u4e0d\u4f9d\u8d56\u4efb\u4f55\u89d2\u5ea6\u5355\u4f4d\u3002<\/p>\n<h4>2.4 \u6b63\u4ea4\u7684\u4e0d\u53d8\u91cf\u523b\u753b&#xff1a;\u5185\u79ef\u4e3a\u96f6<\/h4>\n<p>\u6b63\u4ea4\u53ef\u5b8c\u5168\u4e0d\u501f\u52a9&#034;\u89d2\u5ea6&#034;\u6982\u5ff5\u800c\u5b9a\u4e49\u3002\u7531\u6781\u5316\u6052\u7b49\u5f0f&#xff1a;<\/p>\n<p>u \u00b7 v &#061; (|u&#043;v|\u00b2 \u2212 |u\u2212v|\u00b2) \/ 4<\/p>\n<p>\u8be5\u5b9a\u4e49\u4e2d\u4e0d\u542b\u89d2\u5ea6\u3001\u4e0d\u542b\u5ea6\u3001\u4e0d\u542b\u4efb\u4f55\u5355\u4f4d\u3002 \u4e8e\u662f\u5224\u636e 2.2 \u662f\u4e00\u4e2a\u7eaf\u4ee3\u6570\u7684\u3001\u4e0e\u5ea6\u5236\u65e0\u5173\u7684\u4e0d\u53d8\u91cf\u9648\u8ff0\u3002<\/p>\n<p>\u5176\u7b49\u4ef7\u94fe\u6761\u4e3a&#xff1a;<\/p>\n<p>u \u00b7 v &#061; 0\u3000\u27fa\u3000\u6295\u5f71 proj_v u &#061; 0\u3000\u27fa\u3000u \u4e0d\u542b v \u7684\u6210\u5206\u3000\u27fa\u3000u \u4e0e v \u72ec\u7acb<\/p>\n<p>u \u00b7 v \u2260 0\u3000\u27fa\u3000\u6295\u5f71\u975e\u96f6\u3000\u27fa\u3000\u6210\u5206\u91cd\u53e0\u3000\u27fa\u3000\u4e0d\u72ec\u7acb\u3000\u27fa\u3000\u4e0d\u662f\u65b0\u7ef4\u5ea6<\/p>\n<p>\u6ce8 2.5\u3000\u8be5\u5224\u636e\u4e0d\u662f\u8fd1\u4f3c\u6807\u51c6&#xff1a;\u4efb\u4f55\u5bf9\u6b63\u4ea4\u7684\u504f\u79bb&#xff0c;\u65e0\u8bba\u591a\u5c0f&#xff0c;\u90fd\u610f\u5473\u7740\u8be5\u65b9\u5411\u53ef\u88ab\u65e2\u6709\u65b9\u5411\u7ebf\u6027\u8868\u793a&#xff0c;\u4ece\u800c\u4e27\u5931\u4f5c\u4e3a\u72ec\u7acb\u7ef4\u5ea6\u7684\u8d44\u683c\u3002<\/p>\n<p>\u8bb0\u53f7\u7ea6\u5b9a&#xff1a;\u4e0b\u6587 {e\u2081, e\u2082, e\u2083} \u6052\u8868\u793a R\u00b3 \u7684\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa&#xff1b;\u5411\u91cf\u9ed8\u8ba4\u4e3a\u5355\u4f4d\u5411\u91cf&#xff0c;\u9664\u975e\u53e6\u4f5c\u8bf4\u660e\u3002<\/p>\n<hr \/>\n<h3>3 \u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406<\/h3>\n<h4>3.1 \u5f15\u7406&#xff1a;\u6b63\u4ea4\u7ec4\u7ebf\u6027\u65e0\u5173<\/h4>\n<p>\u5f15\u7406 3.1\u3000\u8bbe v\u2081, \u2026, v_k \u662f\u5185\u79ef\u7a7a\u95f4 V \u4e2d\u4e24\u4e24\u6b63\u4ea4\u7684\u975e\u96f6\u5411\u91cf&#xff0c;\u5219 v\u2081, \u2026, v_k \u7ebf\u6027\u65e0\u5173\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u8bbe \u03a3\u1d62\u208c\u2081\u1d4f c\u1d62 v\u1d62 &#061; 0\u3002\u5bf9\u4efb\u610f\u56fa\u5b9a\u7684 j&#xff0c;\u4e24\u8fb9\u4e0e v_j \u4f5c\u5185\u79ef&#xff1a;<\/p>\n<p>\u03a3\u1d62 c\u1d62 (v\u1d62 \u00b7 v_j) &#061; 0<\/p>\n<p>\u7531\u4e24\u4e24\u6b63\u4ea4\u6027&#xff0c;i \u2260 j \u65f6 v\u1d62 \u00b7 v_j &#061; 0&#xff0c;\u6545\u4e0a\u5f0f\u5316\u4e3a<\/p>\n<p>c_j |v_j|\u00b2 &#061; 0<\/p>\n<p>\u56e0 v_j \u2260 0&#xff0c;\u6709 |v_j|\u00b2 \u2260 0&#xff0c;\u6545 c_j &#061; 0\u3002\u7531 j \u7684\u4efb\u610f\u6027&#xff0c;\u4e00\u5207 c\u1d62 &#061; 0\u3002\u6545 v\u2081, \u2026, v_k \u7ebf\u6027\u65e0\u5173\u3002\u220e<\/p>\n<h4>3.2 \u5b9a\u7406 1&#xff1a;\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406<\/h4>\n<p>\u5b9a\u7406 3.2&#xff08;\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406&#xff09;\u3000R^n \u4e2d\u4e24\u4e24\u6b63\u4ea4\u7684\u975e\u96f6\u65b9\u5411&#xff0c;\u6700\u591a\u6709 n \u4e2a\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u8bbe v\u2081, \u2026, v_k \u4e24\u4e24\u6b63\u4ea4\u4e14\u975e\u96f6\u3002\u7531\u5f15\u7406 3.1&#xff0c;\u5b83\u4eec\u7ebf\u6027\u65e0\u5173\u3002\u800c dim R^n &#061; n&#xff0c;\u6545 k \u2264 n\u3002\u220e<\/p>\n<p>\u63a8\u8bba 3.3\u3000R\u00b3 \u4e2d\u4e24\u4e24\u6b63\u4ea4\u7684\u975e\u96f6\u65b9\u5411&#xff0c;\u6700\u591a\u6709 3 \u4e2a\u3002<\/p>\n<p>\u6ce8 3.4\u3000\u82e5\u628a\u6761\u4ef6\u653e\u5bbd\u4e3a&#034;\u4e0d\u9510&#034;&#xff08;\u5141\u8bb8\u949d\u89d2&#xff0c;\u5373\u5185\u79ef \u2264 0&#xff09;&#xff0c;\u5219 R^n \u4e2d\u6700\u591a\u53ef\u5bb9\u7eb3 2n \u4e2a\u65b9\u5411&#xff08;\u53d6 \u00b1e\u2081, \u2026, \u00b1e_n&#xff09;\u3002\u8fd9\u6070\u6070\u53cd\u8bc1\u4e86\u4e25\u683c\u6b63\u4ea4\u7684\u7a00\u7f3a\u6027&#xff1a;\u5373\u4fbf\u653e\u677e\u6761\u4ef6&#xff0c;\u5bb9\u91cf\u4e5f\u4ec5\u662f\u6709\u9650\u7684 2n&#xff0c;\u800c\u975e\u65e0\u7a77\u3002<\/p>\n<h4>3.3 \u65b9\u5411\u4f59\u5f26\u5b88\u6052\u5f8b<\/h4>\n<p>\u5b9a\u7406 3.5&#xff08;\u65b9\u5411\u4f59\u5f26\u5b88\u6052\u5f8b&#xff09;\u3000\u8bbe u \u662f R^n \u4e2d\u7684\u5355\u4f4d\u5411\u91cf&#xff0c;{e\u2081, \u2026, e_n} \u662f\u4e00\u7ec4\u6b63\u4ea4\u57fa&#xff0c;\u8bb0 \u03b8\u1d62 \u4e3a u \u4e0e e\u1d62 \u7684\u5939\u89d2\u3002\u5219<\/p>\n<p>\u03a3\u1d62\u208c\u2081\u207f cos\u00b2\u03b8\u1d62 &#061; 1<\/p>\n<p>\u8bc1\u660e\u3000\u5c06 u \u5728\u6b63\u4ea4\u57fa\u4e0b\u5c55\u5f00&#xff1a;u &#061; \u03a3\u1d62 (u \u00b7 e\u1d62) e\u1d62\u3002\u7531 |u| &#061; 1 \u53ca Parseval \u6052\u7b49\u5f0f&#xff1a;<\/p>\n<p>1 &#061; |u|\u00b2 &#061; \u03a3\u1d62 (u \u00b7 e\u1d62)\u00b2 &#061; \u03a3\u1d62 cos\u00b2\u03b8\u1d62\u3000\u220e<\/p>\n<p>\u89e3\u91ca\u3000\u6b63\u4ea4\u6027\u662f\u4e00\u79cd\u603b\u91cf\u56fa\u5b9a\u4e3a 1 \u7684&#034;\u9884\u7b97&#034;&#xff0c;\u88ab n \u4e2a\u57fa\u65b9\u5411\u5206\u5149\u3002 \u4e00\u4e2a\u65b9\u5411\u82e5\u8981\u4e0e\u5168\u90e8 n \u4e2a\u57fa\u65b9\u5411\u90fd\u6b63\u4ea4&#xff0c;\u5c31\u5fc5\u987b\u4f7f\u6240\u6709 cos\u00b2\u03b8\u1d62 &#061; 0&#xff0c;\u4ece\u800c \u03a3 cos\u00b2\u03b8\u1d62 &#061; 0 \u2260 1&#xff0c;\u4e0e\u5b88\u6052\u5f8b\u77db\u76fe\u3002<\/p>\n<p>\u6ce8 3.6\u3000\u6ce8\u610f\u4f59\u5f26\u662f\u6bd4\u503c&#xff0c;\u6bd4\u503c\u4e0d\u968f\u89d2\u5ea6\u5355\u4f4d\u6539\u53d8\u3002\u6545\u5b88\u6052\u5f8b\u672c\u8eab\u4e5f\u662f\u5ea6\u5236\u65e0\u5173\u7684\u3002<\/p>\n<h4>3.4 \u5b9a\u7406 2&#xff1a;\u6b63\u4ea4\u4e8f\u635f\u4e0b\u754c<\/h4>\n<p>\u5b9a\u7406 3.7&#xff08;\u6b63\u4ea4\u4e8f\u635f\u4e0b\u754c\u5b9a\u7406&#xff09;\u3000\u5728 R^n \u4e2d&#xff0c;\u4efb\u53d6\u5355\u4f4d\u5411\u91cf u&#xff0c;\u5fc5\u5b58\u5728\u67d0\u4e2a\u57fa\u65b9\u5411 e\u1d62&#xff0c;\u4f7f\u5f97<\/p>\n<p>|cos \u03b8\u1d62| \u2265 1 \/ \u221an<\/p>\n<p>\u7b49\u4ef7\u4e8e&#xff1a;u \u4e0e e\u1d62 \u7684\u5185\u79ef\u7edd\u5bf9\u503c\u4e0d\u5c0f\u4e8e 1\/\u221an&#xff0c;\u5373 u \u5728 e\u1d62 \u4e0a\u7684\u6295\u5f71\u4e0d\u53ef\u5ffd\u7565\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u7531\u5b88\u6052\u5f8b \u03a3\u1d62 cos\u00b2\u03b8\u1d62 &#061; 1 \u53ca\u9e3d\u7b3c\u539f\u7406&#xff0c;\u5b58\u5728 i \u4f7f cos\u00b2\u03b8\u1d62 \u2265 1\/n&#xff0c;\u5373 |cos \u03b8\u1d62| \u2265 1\/\u221an\u3002\u220e<\/p>\n<p>\u63a8\u8bba 3.8&#xff08;\u4e09\u7ef4\u60c5\u5f62&#xff09;\u3000\u5728 R\u00b3 \u4e2d&#xff0c;\u4efb\u4f55\u65b9\u5411\u81f3\u5c11\u4e0e\u67d0\u4e00\u57fa\u65b9\u5411\u6ee1\u8db3 |cos \u03b8| \u2265 1\/\u221a3 \u2248 0.5774&#xff0c;\u5373 |u \u00b7 e\u1d62| \u2265 0.5774\u3002<\/p>\n<p>&#xff08;\u5728 360 \u5ea6\u5236\u4e0b&#xff0c;\u8fd9\u8bfb\u4f5c\u504f\u79bb\u4e0d\u5c0f\u4e8e 35.26\u00b0&#xff1b;\u4f46\u5224\u636e\u672c\u8eab\u662f\u5185\u79ef |u \u00b7 e\u1d62| \u2265 1\/\u221a3&#xff0c;\u4e0e\u5ea6\u5236\u65e0\u5173\u3002&#xff09;<\/p>\n<p>\u6ce8 3.9\u3000\u4e8f\u635f\u4e0b\u754c\u968f n \u589e\u5927\u800c\u5355\u8c03\u51cf\u5c0f&#xff08;n &#061; 4 \u65f6\u4e3a 1\/2&#xff0c;n \u2192 \u221e \u65f6\u8d8b\u4e8e 0&#xff09;\u3002\u8fd9\u8bf4\u660e&#xff1a;\u7ef4\u5ea6\u8d8a\u9ad8&#xff0c;\u7a7a\u95f4\u8d8a&#034;\u677e\u5f1b&#034;\u2014\u2014\u5b83\u4e0d\u518d\u9700\u8981\u4e25\u683c\u6b63\u4ea4\u5c31\u80fd\u5bb9\u7eb3\u66f4\u591a\u65b9\u5411\u3002 \u677e\u5f1b\u6070\u662f\u9ad8\u7ef4\u4e27\u5931\u521a\u6027\u7684\u8868\u73b0&#xff0c;\u8fd9\u4e00\u89c2\u5bdf\u5c06\u5728\u7b2c 6 \u8282\u83b7\u5f97\u62d3\u6251\u5b66\u652f\u6301\u3002<\/p>\n<hr \/>\n<h3>4 \u65b9\u5411\u7684\u65e0\u7a77\u6027\u4e0e&#034;\u4e09\u7ef4\u5305\u542b\u65e0\u7a77\u7ef4&#034;<\/h3>\n<h4>4.1 \u5b9a\u7406 3&#xff1a;\u65b9\u5411\u7684\u8fde\u7eed\u7edf<\/h4>\n<p>\u5b9a\u7406 4.1&#xff08;\u65b9\u5411\u65e0\u7a77\u6027&#xff09;\u3000R\u00b3 \u4e2d\u8fc7\u539f\u70b9\u7684\u65b9\u5411\u6709\u65e0\u7a77\u591a\u4e2a&#xff0c;\u5176\u57fa\u6570\u7b49\u4e8e\u8fde\u7eed\u7edf\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u65b9\u5411\u7684\u96c6\u5408\u5373\u5b9e\u5c04\u5f71\u5e73\u9762 RP\u00b2 &#061; (R\u00b3 \u2216 {0}) \/ ~&#xff0c;\u5176\u4e2d x ~ \u03bbx&#xff08;\u03bb \u2260 0&#xff09;\u3002RP\u00b2 \u2245 S\u00b2 \/ {\u00b11}&#xff0c;\u800c S\u00b2 \u7684\u57fa\u6570\u4e3a\u8fde\u7eed\u7edf&#xff0c;\u5546\u6620\u5c04\u4e0d\u589e\u52a0\u57fa\u6570&#xff0c;\u6545\u65b9\u5411\u96c6\u4e3a\u8fde\u7eed\u7edf\u65e0\u7a77\u3002\u220e<\/p>\n<p>\u56e0\u6b64&#xff1a;\u4efb\u4f55\u53ef\u4ee5\u88ab\u6307\u51fa\u7684&#034;\u7b2c 4 \u65b9\u5411\u3001\u7b2c 5 \u65b9\u5411\u2026\u2026\u7b2c N \u65b9\u5411&#034;&#xff0c;\u4f5c\u4e3a\u4e00\u6761\u65b9\u5411&#xff0c;\u5b83\u65e9\u5df2\u5b58\u5728\u4e8e\u4e09\u7ef4\u4e4b\u4e2d\u3002<\/p>\n<h4>4.2 &#034;\u4e09\u7ef4\u5305\u542b\u65e0\u7a77\u7ef4&#034;\u7684\u786e\u5207\u542b\u4e49<\/h4>\n<p>\u7efc\u5408\u63a8\u8bba 3.3 \u4e0e\u5b9a\u7406 4.1&#xff1a;<\/p>\n<p>\u4e09\u7ef4\u4e2d\u5305\u542b\u65e0\u7a77\u591a\u6761\u65b9\u5411&#xff0c;\u4f46\u5176\u4e2d\u80fd\u4e24\u4e24\u6b63\u4ea4\u7684&#xff0c;\u81f3\u591a\u4e09\u6761\u3002<\/p>\n<p>\u8fd9\u5c31\u662f&#034;\u4e09\u7ef4\u5df2\u7ecf\u5305\u542b\u65e0\u7a77\u7ef4&#034;\u7684\u4e25\u683c\u542b\u4e49&#xff1a;<\/p>\n<ul>\n<li>\u5b83\u4e0d\u662f\u8bf4\u4e09\u7ef4\u88c5\u4e0d\u4e0b\u66f4\u591a\u65b9\u5411\u2014\u2014\u6070\u6070\u76f8\u53cd&#xff0c;\u5b83\u88c5\u4e0b\u4e86\u5168\u90e8\u65b9\u5411&#xff1b;<\/li>\n<li>\u5b83\u662f\u8bf4&#xff1a;\u5728\u8fd9\u65e0\u7a77\u591a\u6761\u65b9\u5411\u4e4b\u4e2d&#xff0c;\u53ea\u6709\u4e09\u6761\u80fd\u540c\u65f6\u6ee1\u8db3\u6b63\u4ea4\u5224\u636e\u3002<\/li>\n<\/ul>\n<p>\u7ed3\u6784\u6027\u5f20\u529b&#xff1a;<\/p>\n<p>\u65b9\u5411\u7684\u6570\u91cf &#061; \u65e0\u7a77\u3000\u4f46\u3000\u6b63\u4ea4\u7684\u5bb9\u91cf &#061; 3<\/p>\n<h4>4.3 \u5b9a\u7406 4&#xff1a;\u5185\u5207\u9000\u5316\u5b9a\u7406<\/h4>\n<p>\u5b9a\u7406 4.2&#xff08;\u5185\u5207\u9000\u5316\u5b9a\u7406&#xff09;\u3000\u8bbe {e\u2081, e\u2082, e\u2083} \u662f R\u00b3 \u7684\u6b63\u4ea4\u57fa\u3002\u4efb\u53d6\u975e\u96f6\u65b9\u5411 u&#xff08;\u5373\u6240\u8c13&#034;\u7b2c\u56db\u65b9\u5411&#034;&#xff09;&#xff0c;\u5219&#xff1a;<\/p>\n<li>\u5b58\u5728 i \u2208 {1, 2, 3}&#xff0c;\u4f7f\u5f97 u \u00b7 e\u1d62 \u2260 0&#xff0c;\u5373 u \u4e0e e\u1d62 \u4e0d\u6b63\u4ea4&#xff1b;<\/li>\n<li>u \u2208 span{e\u2081, e\u2082, e\u2083}&#xff0c;\u5373 u \u662f\u65e2\u6709\u57fa\u7684\u7ebf\u6027\u7ec4\u5408\u3002<\/li>\n<p>\u56e0\u6b64&#xff0c;u \u4e0d\u6784\u6210\u65b0\u7ef4\u5ea6&#xff0c;\u800c\u5305\u542b\u4e8e\u4e09\u7ef4\u4e4b\u4e2d\u3002<\/p>\n<p>\u8bc1\u660e<\/p>\n<p>(1) \u53cd\u8bbe u \u4e0e e\u2081, e\u2082, e\u2083 \u5747\u6b63\u4ea4&#xff0c;\u5373 u \u00b7 e\u1d62 &#061; 0&#xff08;i &#061; 1, 2, 3&#xff09;\u3002\u5c06 u \u5728\u57fa\u4e0b\u5c55\u5f00&#xff1a;<\/p>\n<p>u &#061; (u \u00b7 e\u2081) e\u2081 &#043; (u \u00b7 e\u2082) e\u2082 &#043; (u \u00b7 e\u2083) e\u2083 &#061; 0<\/p>\n<p>\u4e0e u \u2260 0 \u77db\u76fe\u3002\u6545\u81f3\u5c11\u5b58\u5728 i \u4f7f u \u00b7 e\u1d62 \u2260 0\u3002<\/p>\n<p>&#xff08;\u7b49\u4ef7\u5730&#xff0c;\u7531\u5b9a\u7406 3.2&#xff1a;\u82e5 u \u4e0e\u4e09\u4e2a\u6b63\u4ea4\u57fa\u5747\u6b63\u4ea4&#xff0c;\u5219 {e\u2081, e\u2082, e\u2083, u} \u6784\u6210\u56db\u4e2a\u4e24\u4e24\u6b63\u4ea4\u7684\u975e\u96f6\u65b9\u5411&#xff0c;\u4e0e R\u00b3 \u7684\u6b63\u4ea4\u5bb9\u91cf\u4e3a 3 \u77db\u76fe\u3002&#xff09;<\/p>\n<p>(2) \u7531\u6b63\u4ea4\u57fa\u7684\u5b8c\u5907\u6027&#xff0c;u &#061; \u03a3\u1d62 (u \u00b7 e\u1d62) e\u1d62&#xff0c;\u6545 u \u2208 span{e\u2081, e\u2082, e\u2083}\u3002<\/p>\n<p>\u7531 (2) \u77e5 u \u4e0d\u72ec\u7acb&#xff0c;\u4e0d\u643a\u5e26\u65b0\u7684\u72ec\u7acb\u4fe1\u606f&#xff1b;\u7531 (1) \u77e5 u \u4e0e\u65e2\u6709\u57fa\u975e\u6b63\u4ea4&#xff0c;\u6545\u6309\u5224\u636e 2.2 \u4e0d\u5177\u5907\u65b0\u7ef4\u5ea6\u8d44\u683c\u3002\u220e<\/p>\n<p>\u63a8\u8bba 4.3&#xff08;\u6b63\u4ea4\u8868\u8ff0&#xff09;\u3000\u7528\u738b\u78ca\u5224\u636e\u7684\u8bed\u8a00\u8868\u8ff0&#xff0c;\u5b9a\u7406 4.2 \u5373&#xff1a;<\/p>\n<p>\u5f53\u628a\u4e00\u4e2a\u989d\u5916\u7684\u7ef4\u5ea6&#034;\u5185\u5207&#034;\u8fdb\u4e09\u7ef4\u73b0\u5b9e\u65f6&#xff0c;\u5b83\u4e0e\u65e2\u6709\u65b9\u5411\u4e0d\u518d\u6b63\u4ea4\u3002<\/p>\n<p>&#xff08;\u5728 360 \u5ea6\u5236\u4e0b&#xff0c;\u8fd9\u8bfb\u4f5c&#034;\u90a3\u4e2a\u89d2\u5df2\u7ecf\u4e0d\u662f 90 \u5ea6\u4e86&#034;\u3002\u4f46\u672c\u8d28\u662f\u5185\u79ef\u975e\u96f6&#xff0c;\u4e0e\u5ea6\u5236\u65e0\u5173\u3002&#xff09;<\/p>\n<p>\u8fd9\u6b63\u662f\u7ef4\u5ea6\u6784\u9020\u5728\u4e09\u7ef4\u4e2d\u4e0d\u53ef\u5b9e\u73b0\u7684\u51e0\u4f55\u6839\u6e90\u3002<\/p>\n<h4>4.4 \u9ad8\u7ef4\u7684\u4e24\u6761\u8def\u5f84\u53ca\u5176\u7a77\u5c3d<\/h4>\n<table>\n<tr>\u8def\u5f84\u7ed3\u679c\u5224\u5b9a<\/tr>\n<tbody>\n<tr>\n<td>\u5185\u5207&#xff1a;\u4f5c\u4e3a\u4e09\u7ef4\u4e2d\u7684\u4e00\u6761\u65b9\u5411<\/td>\n<td>\u4e0e\u67d0\u57fa\u65b9\u5411\u5185\u79ef\u975e\u96f6&#xff08;\u5b9a\u7406 4.2&#xff09;&#xff1b;\u9000\u5316\u4e3a\u7ebf\u6027\u7ec4\u5408<\/td>\n<td>\u4e0d\u662f\u65b0\u7ef4\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>\u5916\u7f6e&#xff1a;\u4f5c\u4e3a R\u2074 \u7684\u4e00\u90e8\u5206\u7559\u5728\u4e09\u7ef4\u4e4b\u5916<\/td>\n<td>\u4e0d\u5728\u4e09\u7ef4\u4e4b\u4e2d&#xff0c;\u65e0\u6cd5\u88ab\u6784\u9020\u3001\u65e0\u6cd5\u88ab\u6d4b\u91cf\u3001\u65e0\u6cd5\u88ab\u4f5c\u7528<\/td>\n<td>\u4e0d\u53ef\u6784\u9020<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u4e24\u6761\u8def\u5f84\u5747\u5df2\u7a77\u5c3d&#xff0c;\u4e14\u5747\u4e0d\u901a\u5411&#034;\u5b58\u5728&#034;\u3002\u6545&#xff1a;<\/p>\n<p>\u63a8\u8bba 4.4&#xff08;\u9ad8\u7ef4\u4e0d\u53ef\u6784\u9020\u6027&#xff09;\u3000\u5728\u4e09\u7ef4\u65e2\u5b9a\u821e\u53f0\u5185&#xff0c;\u4e00\u5207 n \u2265 4 \u7684\u7ef4\u5ea6\u5747\u4e0d\u53ef\u6784\u9020\u3002<\/p>\n<hr \/>\n<h3>5 \u4f4e\u7ef4\u7684\u9000\u5316\u6027<\/h3>\n<h4>5.1 \u4e00\u7ef4&#xff1a;\u6b63\u4ea4\u5173\u7cfb\u7684\u4e8c\u5143\u6027\u77db\u76fe<\/h4>\n<p>\u547d\u9898 5.1\u3000\u4e00\u7ef4\u4e0d\u6784\u6210\u81ea\u6d3d\u7684\u7ef4\u5ea6\u7cfb\u7edf\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u6b63\u4ea4\u662f\u4e8c\u5143\u5173\u7cfb&#xff1a;u \u22a5 v \u9700\u8981\u4e24\u4e2a\u65b9\u5411\u624d\u80fd\u88ab\u5efa\u7acb\u3002\u5728 R\u00b9 \u4e2d&#xff0c;\u8fc7\u539f\u70b9\u53ea\u6709\u4e00\u4e2a\u65b9\u5411\u3002\u7531\u4e8e\u4e0d\u5b58\u5728\u7b2c\u4e8c\u4e2a\u65b9\u5411\u4e0e\u4e4b\u6784\u6210\u5173\u7cfb&#xff0c;\u6b63\u4ea4\u5173\u7cfb\u65e0\u4ece\u5efa\u7acb\u3002<\/p>\n<p>\u6309\u5b9a\u4e49 2.1&#xff0c;\u7ef4\u5ea6\u7531\u4e24\u4e24\u6b63\u4ea4\u65b9\u5411\u7684\u6700\u5927\u6570\u76ee\u523b\u753b&#xff1b;\u800c\u4e00\u7ef4\u4e2d\u8be5\u6570\u76ee\u65e0\u6cd5\u8d85\u8fc7 1&#xff0c;\u4e14\u4ec5\u6709\u7684\u8fd9 1 \u4e2a\u65b9\u5411\u4e5f\u56e0\u7f3a\u4e4f\u53c2\u7167\u800c\u65e0\u6cd5\u88ab\u786e\u8ba4\u4e3a&#034;\u6b63\u4ea4\u65b9\u5411&#034;\u3002<\/p>\n<p>\u4ee3\u6570\u4e0a\u770b&#xff0c;\u4e00\u7ef4\u65cb\u8f6c\u7fa4 SO(1) &#061; {1} \u662f\u5e73\u51e1\u7fa4&#xff1a;\u4e0d\u5b58\u5728\u4efb\u4f55\u975e\u5e73\u51e1\u7684\u8f6c\u52a8\u81ea\u7531\u5ea6&#xff0c;\u4e00\u4e2a\u65b9\u5411\u65e0\u6cd5\u88ab\u8f6c\u52a8\u5230\u4efb\u4f55\u72ec\u7acb\u7684\u5176\u4ed6\u4f4d\u7f6e\u3002\u6545\u4e00\u7ef4\u662f\u9000\u5316\u7684\u3002\u220e<\/p>\n<p>\u6ce8 5.2\u3000\u4e00\u7ef4\u53ef\u8868\u793a\u4e3a R\u00b3 \u4e2d\u9000\u5316\u7684\u7ebf&#xff1a;L &#061; {(x, 0, 0)}\u3002\u5b83\u662f\u4e09\u7ef4\u7684\u4e00\u7ef4\u622a\u9762&#xff0c;\u800c\u975e\u81ea\u8db3\u7a7a\u95f4\u3002<\/p>\n<h4>5.2 \u4e8c\u7ef4&#xff1a;\u5185\u5916\u4e0d\u53ef\u754c\u5b9a<\/h4>\n<p>\u547d\u9898 5.3\u3000\u4e8c\u7ef4\u4e0d\u6784\u6210\u81ea\u8db3\u7684\u7ef4\u5ea6\u7cfb\u7edf\u3002<\/p>\n<p>\u8bc1\u660e\u3000\u4e8c\u7ef4\u4e2d\u53ef\u6709\u4e00\u5bf9\u6b63\u4ea4\u65b9\u5411 e\u2081 \u22a5 e\u2082 \u2713\u3002\u4f46\u8981\u6210\u4e3a\u4e00\u4e2a\u81ea\u8db3\u7684\u4e16\u754c&#xff0c;\u5b83\u5fc5\u987b\u80fd\u591f\u754c\u5b9a&#034;\u5e73\u9762\u4e4b\u5185&#034;\u4e0e&#034;\u5e73\u9762\u4e4b\u5916&#034;\u3002<\/p>\n<p>\u800c&#034;\u5185&#xff0f;\u5916&#034;\u7684\u533a\u5206&#xff0c;\u9700\u8981\u4e00\u4e2a\u4e0d\u5728\u8be5\u5e73\u9762\u5185\u7684\u89c6\u89d2\u3002\u8bbe\u89c2\u5bdf\u8005\u5728\u5e73\u9762\u5185&#xff0c;\u5219\u4ed6\u7684\u4e00\u5207\u64cd\u4f5c\u4e0e\u89c2\u6d4b\u5747\u88ab\u9650\u5236\u5728 span{e\u2081, e\u2082} \u4e2d&#xff0c;\u65e0\u6cd5\u786e\u8ba4\u8be5\u7a7a\u95f4\u662f\u5426&#034;\u5b8c\u6574&#034;&#xff0c;\u4e5f\u65e0\u6cd5\u786e\u8ba4\u81ea\u8eab\u7684\u89c2\u6d4b\u662f\u5426\u7a77\u5c3d\u3002<\/p>\n<p>\u5f62\u5f0f\u5730&#xff1a;\u4e8c\u7ef4\u7684\u6cd5\u65b9\u5411 n &#061; e\u2081 \u00d7 e\u2082 \u4e0d\u5728\u8be5\u7a7a\u95f4\u5185&#xff0c;\u56e0\u6b64\u8be5\u7a7a\u95f4\u5728\u81ea\u8eab\u5185\u90e8\u65e0\u6cd5\u5b9a\u4e49\u81ea\u8eab\u7684\u8fb9\u754c\u3002\u6545\u4e8c\u7ef4\u5fc5\u987b\u88ab\u5d4c\u5165\u4e09\u7ef4\u624d\u80fd\u88ab\u8c08\u8bba\u2014\u2014\u5b83\u662f\u4e09\u7ef4\u7684\u4e8c\u7ef4\u622a\u9762&#xff0c;\u800c\u975e\u81ea\u8db3\u7a7a\u95f4\u3002\u220e<\/p>\n<p>\u6ce8 5.4&#xff08;\u7269\u7406\u4fa7\u4f50\u8bc1&#xff09;\u3000\u82e5\u8981\u6c42\u4e8c\u7ef4\u771f\u6b63\u81ea\u8db3&#xff0c;\u5176\u539a\u5ea6\u987b\u4e25\u683c\u4e3a\u96f6&#xff0c;\u5373 \u0394z &#061; 0\u3002\u7531\u4e0d\u786e\u5b9a\u6027\u539f\u7406 \u0394z \u00b7 \u0394p_z \u2265 \u0127\/2&#xff0c;\u5f97 \u0394p_z \u2192 \u221e&#xff0c;\u80fd\u91cf\u53d1\u6563\u3002\u6545\u96f6\u539a\u5ea6\u7684\u4e8c\u7ef4\u5728\u7269\u7406\u4e0a\u4e0d\u53ef\u5b9e\u73b0\u3002\u6b64\u5c5e\u7269\u7406\u5c42\u6b21&#xff0c;\u4ec5\u4f5c\u4f50\u8bc1\u3002<\/p>\n<h4>5.3 \u5b9a\u7406 5&#xff1a;\u4f4e\u7ef4\u9000\u5316\u5b9a\u7406<\/h4>\n<p>\u5b9a\u7406 5.5\u3000n &#061; 1 \u4e0e n &#061; 2 \u5747\u4e0d\u53ef\u6784\u9020\u4e3a\u81ea\u8db3\u7684\u7ef4\u5ea6\u7cfb\u7edf\u3002\u4e8c\u8005\u5206\u522b\u662f R\u00b3 \u7684\u4e00\u7ef4\u622a\u9762\u4e0e\u4e8c\u7ef4\u622a\u9762&#xff1a;<\/p>\n<p>R\u00b9 \u2245 span{e\u2081} \u2282 R\u00b3&#xff0c;\u3000R\u00b2 \u2245 span{e\u2081, e\u2082} \u2282 R\u00b3<\/p>\n<hr \/>\n<h3>6 \u62d3\u6251\u5b66\u8bc1\u636e&#xff1a;\u4e09\u7ef4\u7684\u552f\u4e00\u521a\u6027<\/h3>\n<h4>6.1 \u8bc1\u636e\u4e00&#xff1a;\u7ebd\u7ed3\u2014\u2014\u7ea0\u7f20\u53ea\u5728\u4e09\u7ef4\u5b58\u5728<\/h4>\n<p>\u4e8b\u5b9e 6.1&#xff08;\u7ebd\u7ed3\u7684\u7ef4\u5ea6\u552f\u4e00\u6027&#xff09;\u3000S\u00b9 \u5728 R^n \u4e2d\u7684\u5d4c\u5165&#xff08;\u7ebd\u7ed3&#xff09;&#xff1a;<\/p>\n<table>\n<tr>\u7ef4\u5ea6\u7ebd\u7ed3\u60c5\u51b5<\/tr>\n<tbody>\n<tr>\n<td>n &#061; 1<\/td>\n<td>\u65e0\u6cd5\u5d4c\u5165\u95ed\u5408\u66f2\u7ebf<\/td>\n<\/tr>\n<tr>\n<td>n &#061; 2<\/td>\n<td>\u7531 Jordan \u66f2\u7ebf\u5b9a\u7406&#xff0c;\u5e73\u9762\u95ed\u66f2\u7ebf\u65e0\u7ebd\u7ed3<\/td>\n<\/tr>\n<tr>\n<td>n &#061; 3<\/td>\n<td>\u5b58\u5728\u65e0\u7a77\u591a\u4e2a\u4e92\u4e0d\u7b49\u4ef7\u7684\u3001\u4e0d\u53ef\u89e3\u5f00\u7684\u975e\u5e73\u51e1\u7ebd\u7ed3&#xff08;\u4e09\u53f6\u7ed3\u30018 \u5b57\u7ed3\u2026\u2026&#xff09;&#xff0c;\u5176\u57fa\u672c\u7fa4 \u03c0\u2081(R\u00b3 \u2216 K) \u975e\u5e73\u51e1<\/td>\n<\/tr>\n<tr>\n<td>n \u2265 4<\/td>\n<td>\u4e00\u5207\u7ebd\u7ed3\u540c\u75d5\u4e8e\u5e73\u51e1\u7ed3&#xff0c;\u5747\u53ef\u88ab\u89e3\u5f00<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u539f\u56e0\u3000\u5728 n \u2265 4 \u4e2d&#xff0c;\u7ebd\u7ed3\u56fe\u7684\u4e00\u4e2a\u4ea4\u53c9\u70b9\u53ef\u901a\u8fc7\u7b2c\u56db\u5750\u6807\u65b9\u5411\u88ab&#034;\u63a8\u5f00&#034;\u2014\u2014\u4f59\u7ef4\u8db3\u591f\u5927&#xff0c;\u4f7f\u5f97\u89e3\u4ea4\u53c9\u7684\u5fae\u6270\u603b\u662f\u53ef\u884c\u3002\u800c\u5728\u4e09\u7ef4\u4e2d&#xff0c;\u4f59\u7ef4\u4e0d\u8db3&#xff0c;\u4ea4\u53c9\u88ab\u9501\u5b9a\u3002<\/p>\n<p>\u610f\u4e49\u3000\u6df7\u6c8c\u4f5c\u4e3a\u672c\u4f53&#xff0c;\u5176\u6839\u672c\u7279\u5f81\u662f\u4e0d\u53ef\u89e3\u5f00\u7684\u7ea0\u7f20\u3002\u800c\u552f\u4e00\u80fd\u627f\u8f7d\u4e0d\u53ef\u89e3\u5f00\u4e4b\u7ea0\u7f20\u7684\u7ef4\u5ea6&#xff0c;\u662f\u4e09\u7ef4\u3002<\/p>\n<p>\u4f4e\u7ef4\u592a\u8d2b\u7620&#xff0c;\u5bb9\u4e0d\u4e0b\u7ea0\u7f20&#xff1b;\u9ad8\u7ef4\u592a\u677e\u5f1b&#xff0c;\u5bb9\u4e0d\u4f4f\u7ea0\u7f20\u3002\u552f\u6709\u4e09\u7ef4&#xff0c;\u6070\u597d\u7d27\u5230\u80fd\u9501\u4f4f\u7ea0\u7f20\u3002<\/p>\n<p>\u8fd9\u662f&#034;\u6df7\u6c8c\u62d3\u6251\u5b66&#034;\u8fd9\u4e00\u547d\u540d\u7684\u76f4\u63a5\u6570\u5b66\u6839\u636e\u3002<\/p>\n<h4>6.2 \u8bc1\u636e\u4e8c&#xff1a;Whitney \u624b\u672f\u7684\u7ef4\u6570\u95e8\u69db<\/h4>\n<p>\u4e8b\u5b9e 6.2\u3000\u5728 n \u7ef4\u6d41\u5f62\u4e0a\u505a\u624b\u672f\u4ee5\u6d88\u9664\u53cc\u70b9\u65f6&#xff0c;Whitney \u6280\u5de7\u8981\u6c42 2 &#043; 2 &lt; n&#xff0c;\u5373 n \u2265 5\u3002<\/p>\n<ul>\n<li>n \u2265 5&#xff1a;Whitney \u6280\u5de7\u76f4\u63a5\u53ef\u7528&#xff0c;\u624b\u672f\u65e0\u969c\u788d&#xff1b;<\/li>\n<li>n &#061; 4&#xff1a;\u9700 Freedman \u7684 Casson handle \u7406\u8bba&#xff08;1982&#xff09;&#xff0c;\u6781\u5176\u56f0\u96be&#xff1b;<\/li>\n<li>n &#061; 3&#xff1a;Whitney \u6280\u5de7\u6839\u672c\u4e0d\u53ef\u7528&#xff0c;\u7a7a\u95f4\u4e0d\u8db3&#xff0c;\u5fc5\u987b\u53e6\u8f9f\u8e4a\u5f84&#xff08;Perelman \u7684 Ricci \u6d41&#xff0c;2003&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u610f\u4e49\u3000\u9ad8\u7ef4\u4e4b\u6240\u4ee5&#034;\u5bb9\u6613\u64cd\u4f5c&#034;&#xff0c;\u6b63\u56e0\u4e3a\u5b83\u677e\u5f1b\u2014\u2014\u6709\u591a\u4f59\u7684\u7ef4\u5ea6\u53ef\u7528\u4f5c\u64cd\u4f5c\u7684\u4f59\u5730\u3002\u4e09\u7ef4\u4e4b\u6240\u4ee5&#034;\u575a\u786c&#034;&#xff0c;\u6b63\u56e0\u4e3a\u5b83\u6ca1\u6709\u4f59\u91cf\u3002\u677e\u5f1b\u610f\u5473\u7740\u7ed3\u6784\u6613\u4e8e\u88ab\u6270\u52a8\u3001\u88ab\u89e3\u5f00\u3001\u88ab\u62b9\u5e73&#xff1b;\u521a\u6027\u610f\u5473\u7740\u7ed3\u6784\u80fd\u4fdd\u6301\u3001\u80fd\u627f\u8f7d\u3001\u80fd\u6210\u4e3a\u5b9e\u5728\u3002<\/p>\n<h4>6.3 \u8bc1\u636e\u4e09&#xff1a;Poincar\u00e9 \u731c\u60f3\u7684\u96be\u5ea6\u8c31<\/h4>\n<p>\u4e8b\u5b9e 6.3\u3000\u5e7f\u4e49 Poincar\u00e9 \u731c\u60f3\u7684\u8bc1\u660e\u5e74\u4ee3&#xff1a;<\/p>\n<table>\n<tr>\u7ef4\u5ea6\u8bc1\u660e\u8005\u5e74\u4ee3<\/tr>\n<tbody>\n<tr>\n<td>n \u2265 5<\/td>\n<td>Smale<\/td>\n<td>1961<\/td>\n<\/tr>\n<tr>\n<td>n &#061; 4<\/td>\n<td>Freedman<\/td>\n<td>1982<\/td>\n<\/tr>\n<tr>\n<td>n &#061; 3<\/td>\n<td>Perelman<\/td>\n<td>2003<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u610f\u4e49\u3000\u4e09\u7ef4\u662f\u6700\u665a\u88ab\u653b\u514b\u7684\u3001\u4e5f\u662f\u6700\u96be\u7684\u3002\u7ef4\u6570\u8d8a\u9ad8\u8d8a\u677e\u5f1b\u56e0\u800c\u8d8a\u7b80\u5355&#xff0c;\u7ef4\u6570\u8d8a\u4f4e\u8d8a\u521a\u6027\u56e0\u800c\u8d8a\u56f0\u96be\u3002 \u4e09\u7ef4\u5904\u5728&#034;\u521a\u6027\u7684\u6781\u503c&#034;\u4e0a\u3002<\/p>\n<h4>6.4 \u8bc1\u636e\u56db&#xff1a;\u53c9\u79ef\u4e0e\u95ed\u5408\u6027<\/h4>\n<p>\u4e8b\u5b9e 6.4\u3000\u53cc\u7ebf\u6027\u3001\u53cd\u5bf9\u79f0\u3001\u4e14\u4e0e\u4e24\u8f93\u5165\u5747\u6b63\u4ea4\u7684\u5411\u91cf\u79ef&#xff08;\u53c9\u79ef&#xff09;&#xff0c;\u4ec5\u5728 n &#061; 3 \u4e0e n &#061; 7 \u5b58\u5728&#xff08;\u5206\u522b\u5bf9\u5e94\u56db\u5143\u6570 H \u4e0e\u516b\u5143\u6570 O \u7684\u865a\u90e8&#xff0c;\u7531 Hurwitz \u5b9a\u7406\u4e0e Adams \u7684 Hopf \u4e0d\u53d8\u91cf\u4e00\u5b9a\u7406\u6240\u9650\u5b9a&#xff09;\u3002<\/p>\n<p>\u610f\u4e49\u3000\u5728\u4e09\u7ef4\u4e2d&#xff0c;\u4e24\u4e2a\u72ec\u7acb\u65b9\u5411\u53ef\u4ee5\u5728\u81ea\u8eab\u7a7a\u95f4\u5185\u90e8\u552f\u4e00\u5730\u786e\u5b9a\u7b2c\u4e09\u4e2a\u6b63\u4ea4\u65b9\u5411&#xff1a;<\/p>\n<p>e\u2081 \u00d7 e\u2082 &#061; e\u2083&#xff0c;\u4e14 e\u2083 \u00b7 e\u2081 &#061; 0&#xff0c;e\u2083 \u00b7 e\u2082 &#061; 0<\/p>\n<p>\u8fd9\u662f&#034;\u7531\u6700\u5c11\u53c2\u7167\u786e\u5b9a\u65b0\u65b9\u5411&#034;\u7684\u95ed\u5408\u7ed3\u6784\u3002\u4e03\u7ef4\u867d\u4e5f\u6709\u7c7b\u4f3c\u7ed3\u6784&#xff0c;\u4f46\u4e0d\u6ee1\u8db3\u4e8b\u5b9e 6.1&#xff08;\u65e0\u7ebd\u7ed3\u9501\u5b9a&#xff09;\u3001\u4e0d\u6ee1\u8db3\u4e8b\u5b9e 6.2&#xff08;\u624b\u672f\u677e\u5f1b&#xff09;&#xff0c;\u4e14\u4e0d\u5177\u5907\u7b2c 7 \u8282\u7684\u7269\u7406\u7a33\u5b9a\u6027\u3002\u6545\u4e09\u7ef4\u662f\u552f\u4e00\u540c\u65f6\u6ee1\u8db3\u5168\u90e8\u6761\u4ef6\u7684\u7ef4\u5ea6\u3002<\/p>\n<h4>6.5 \u5b9a\u7406 6&#xff1a;\u4e34\u754c\u7ef4\u5ea6\u5b9a\u7406<\/h4>\n<p>\u5b9a\u7406 6.5&#xff08;\u4e34\u754c\u7ef4\u5ea6\u5b9a\u7406&#xff09;\u3000\u4e09\u7ef4\u662f\u552f\u4e00\u7684\u4e34\u754c\u7ef4\u5ea6&#xff0c;\u5373\u552f\u4e00\u6ee1\u8db3\u4e0b\u5217\u5168\u90e8\u6761\u4ef6\u7684\u7a7a\u95f4\u7ef4\u5ea6\u6570\u2014\u2014<\/p>\n<li>\u6b63\u4ea4\u57fa\u5b8c\u6574\u4e14\u5df2\u7a77\u5c3d&#xff08;\u63a8\u8bba 3.3&#xff09;&#xff1b;<\/li>\n<li>\u80fd\u627f\u8f7d\u4e0d\u53ef\u89e3\u5f00\u7684\u7ea0\u7f20&#xff08;\u4e8b\u5b9e 6.1&#xff09;&#xff1b;<\/li>\n<li>\u7a7a\u95f4\u65e0\u4f59\u91cf&#xff0c;\u7ed3\u6784\u5177\u521a\u6027&#xff08;\u4e8b\u5b9e 6.2&#xff09;&#xff1b;<\/li>\n<li>\u6b63\u4ea4\u8865\u7684\u6784\u9020\u5728\u7a7a\u95f4\u5185\u90e8\u95ed\u5408&#xff08;\u4e8b\u5b9e 6.4&#xff09;&#xff1b;<\/li>\n<li>\u80fd\u627f\u8f7d\u7a33\u5b9a\u7684\u675f\u7f1a\u7ed3\u6784&#xff08;\u7b2c 7 \u8282&#xff09;\u3002<\/li>\n<hr \/>\n<h3>7 \u7269\u7406\u5c42\u6b21\u7684\u8bc1\u636e<\/h3>\n<h4>7.1 \u9ad8\u65af\u5b9a\u5f8b\u4e0e\u4e2d\u5fc3\u529b<\/h4>\n<p>n \u7ef4\u4e2d\u534a\u5f84\u4e3a r \u7684\u7403\u9762\u9762\u79ef \u221d r^{n\u22121}\u3002\u7531\u901a\u91cf\u5b88\u6052&#xff08;\u9ad8\u65af\u5b9a\u5f8b&#xff09;&#xff1a;<\/p>\n<p>F\u00ae \u221d 1 \/ r^{n\u22121}&#xff0c;\u6545 V\u00ae \u221d \u22121 \/ r^{n\u22122}\u3000&#xff08;n \u2260 2&#xff09;<\/p>\n<h4>7.2 Bertrand \u5b9a\u7406\u7684\u552f\u4e00\u6311\u9009<\/h4>\n<p>Bertrand \u5b9a\u7406&#xff08;1873&#xff09;\u3000\u5728\u4e00\u5207\u4e2d\u5fc3\u52bf\u4e2d&#xff0c;\u53ea\u6709 V \u221d \u22121\/r \u4e0e V \u221d r\u00b2 \u80fd\u4f7f\u6240\u6709\u6709\u754c\u8f68\u9053\u95ed\u5408&#xff08;\u5373\u7a33\u5b9a&#xff09;\u3002<\/p>\n<p>\u4ee4 \u2212(n\u22122) &#061; \u22121&#xff0c;\u5f97 n &#061; 3\u3002<\/p>\n<ul>\n<li>n &#061; 2&#xff1a;V \u221d ln r&#xff0c;\u4e0d\u5728\u540d\u5355\u5185 \u2192 \u8f68\u9053\u5fc5\u8fdb\u52a8\u6216\u9003\u9038&#xff1b;<\/li>\n<li>n \u2265 4&#xff1a;V \u221d \u22121\/r^{n\u22122}&#xff0c;\u4e0d\u5728\u540d\u5355\u5185 \u2192 \u5fae\u6270\u5373\u87ba\u65cb\u5760\u6bc1\u6216\u9003\u9038\u3002<\/li>\n<\/ul>\n<p>\u21d2 \u7269\u7406\u5224\u636e\u5728\u6570\u5b66\u7684\u65e0\u77db\u76fe\u7ef4\u5ea6\u65cf\u4e2d&#xff0c;\u552f\u4e00\u5730\u6311\u51fa n &#061; 3\u3002<\/p>\n<h4>7.3 Ehrenfest \u8bba\u8bc1\u4e0e\u91cf\u5b50\u7a33\u5b9a\u6027<\/h4>\n<p>Ehrenfest&#xff08;1917&#xff09;\u9996\u6b21\u7cfb\u7edf\u63d0\u51fa\u4e0a\u8ff0\u8bba\u8bc1\u3002\u91cf\u5b50\u5c42\u9762\u540c\u5411&#xff1a;n \u2265 4 \u65f6\u6709\u6548\u52bf\u4e2d\u79bb\u5fc3\u9879 \u221d &#043;1\/r\u00b2 \u65e0\u6cd5\u4e0e \u221d \u22121\/r^{n\u22122} \u5e73\u8861&#xff0c;\u539f\u5b50\u65e0\u7a33\u5b9a\u57fa\u6001&#xff1b;n &#061; 2 \u540c\u6837\u9000\u5316\u3002\u21d2 \u53ea\u6709\u4e09\u7ef4\u5141\u8bb8\u7a33\u5b9a\u7684\u675f\u7f1a\u7ed3\u6784&#xff08;\u884c\u661f\u8f68\u9053\u3001\u539f\u5b50&#xff09;\u3002<\/p>\n<h4>7.4 \u4e00\u81f4\u6027<\/h4>\n<p>\u7269\u7406\u5224\u636e&#xff08;\u7a33\u5b9a\u675f\u7f1a\u7ed3\u6784&#xff09;\u3001\u51e0\u4f55\u5224\u636e&#xff08;\u5185\u79ef\u4e3a\u96f6\u7684\u6b63\u4ea4&#xff09;\u3001\u62d3\u6251\u5224\u636e&#xff08;\u7ebd\u7ed3\u9501\u5b9a&#xff09;\u4e09\u8005\u72ec\u7acb&#xff0c;\u5374\u4e00\u81f4\u5730\u9009\u51fa n &#061; 3\u3002\u4e09\u6761\u72ec\u7acb\u8bc1\u636e\u94fe\u7684\u540c\u5f52&#xff0c;\u6784\u6210\u5bf9\u4e09\u7ef4\u552f\u4e00\u6027\u7684\u5f3a\u4ea4\u53c9\u9a8c\u8bc1\u3002<\/p>\n<hr \/>\n<h3>8 \u4e09\u7ef4\u552f\u4e00\u539f\u7406<\/h3>\n<p>\u4e09\u7ef4\u552f\u4e00\u539f\u7406&#xff08;Principle of Three-Dimensional Uniqueness&#xff09;<\/p>\n<p>\u8bbe W \u4e3a\u53ef\u6784\u9020\u7684\u7269\u7406\u7a7a\u95f4\u3002\u5219&#xff1a;<\/p>\n<p>(i) \u9000\u5316\u6027\u3000n &#061; 1 \u4e0e n &#061; 2 \u4e0d\u53ef\u6784\u9020\u4e3a\u81ea\u8db3\u7ef4\u5ea6\u7cfb\u7edf&#xff1b;\u5b83\u4eec\u662f R\u00b3 \u7684\u622a\u9762&#xff1a;R\u00b9 \u2282 R\u00b2 \u2282 R\u00b3\u3002<\/p>\n<p>(ii) \u95ed\u5408\u6027\u3000n &#061; 3 \u662f\u552f\u4e00\u6ee1\u8db3\u6b63\u4ea4\u5224\u636e&#xff08;\u5b9a\u4e49 2.1&#xff09;\u4e14\u81ea\u6d3d\u95ed\u5408\u7684\u7ef4\u5ea6\u6570&#xff1a;\u5176\u6b63\u4ea4\u57fa\u5b8c\u6574&#xff0c;\u4e14\u7531\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406\u77e5\u5df2\u7a77\u5c3d\u3002<\/p>\n<p>(iii) \u5305\u542b\u6027\u3000\u4e00\u5207 n \u2265 4 \u7684\u65b9\u5411&#xff0c;\u82e5\u5185\u5207\u4e8e R\u00b3&#xff0c;\u5219\u5fc5\u4e0e\u67d0\u4e00\u57fa\u65b9\u5411\u975e\u6b63\u4ea4&#xff08;\u5b9a\u7406 4.2&#xff09;&#xff0c;\u9000\u5316\u4e3a\u65e2\u6709\u57fa\u7684\u7ebf\u6027\u7ec4\u5408&#xff0c;\u56e0\u800c\u5305\u542b\u4e8e\u4e09\u7ef4&#xff1b;\u82e5\u4e0d\u5185\u5207&#xff0c;\u5219\u4e0d\u53ef\u6784\u9020\u3002<\/p>\n<p>(iv) \u552f\u4e00\u6027\u3000\u6545\u4e09\u7ef4\u4e0d\u662f\u4f17\u591a\u7ef4\u5ea6\u4e2d\u7684\u4e00\u4e2a&#xff0c;\u800c\u662f\u7ef4\u5ea6\u6982\u5ff5\u5f97\u4ee5\u6210\u7acb\u7684\u552f\u4e00\u821e\u53f0&#xff1a;<\/p>\n<p>\u4f4e\u7ef4\u662f\u4e09\u7ef4\u7684\u9000\u5316\u6295\u5f71&#xff0c;\u9ad8\u7ef4\u662f\u4e09\u7ef4\u7684\u7b26\u53f7\u53e0\u52a0\u3002<\/p>\n<h4>\u63a8\u8bba<\/h4>\n<p>\u63a8\u8bba 8.1&#xff08;\u9ad8\u7ef4\u662f\u63cf\u8ff0\u5c42&#xff0c;\u4e0d\u662f\u7a7a\u95f4&#xff09;\u3000\u6240\u8c13&#034;\u7b2c\u56db\u7ef4\u3001\u7b2c\u4e94\u7ef4\u2026\u2026\u7b2c N \u7ef4&#034;&#xff0c;\u5728\u4e09\u7ef4\u73b0\u5b9e\u4e2d\u53ea\u53ef\u80fd\u662f\u4e09\u7ef4\u5bf9\u8c61\u4e0a\u7684\u51fd\u6570\u5c42\u6216\u53c2\u6570\u5c42&#xff08;\u7ea4\u7ef4\u4e1b\u7684\u7ea4\u7ef4\u3001\u7279\u5f81\u7a7a\u95f4\u3001\u76f8\u7a7a\u95f4\u5750\u6807&#xff09;\u3002\u5b83\u4eec\u662f\u6807\u7b7e&#xff0c;\u4e0d\u662f\u4f4d\u79fb\u65b9\u5411\u2014\u2014\u65e0\u6cd5\u5728\u5176\u4e0a&#034;\u8d70\u4e00\u6b65&#034;\u3002\u4f5c\u4e3a\u63cf\u8ff0\u7ed3\u6784&#xff0c;\u5b83\u4eec\u5efa\u7acb\u5728\u4e09\u7ef4\u4e4b\u4e0a&#xff0c;\u6545\u5305\u542b\u4e8e\u4e09\u7ef4\u7684\u63cf\u8ff0\u4e4b\u4e2d\u3002<\/p>\n<p>\u63a8\u8bba 8.2&#xff08;\u6295\u5f71\u4e0d\u53ef\u9006&#xff09;\u3000\u7269\u7406\u53ef\u89c2\u6d4b\u91cf\u5fc5\u987b\u843d\u5728\u5b9e\u6570\u8c31\u4e0a&#xff0c;\u800c\u7406\u8bba\u63cf\u8ff0\u53ef\u4f7f\u7528\u590d\u6570\u3001\u9ad8\u7ef4 Hilbert \u7a7a\u95f4\u3001\u7ea4\u7ef4\u4e1b\u7b49\u3002\u6545&#034;\u6570\u5b66\u63cf\u8ff0 \u2192 \u7269\u7406\u5b9e\u5728&#034;\u662f\u4e00\u4e2a\u4e0d\u53ef\u9006\u6295\u5f71&#xff0c;\u800c\u975e\u6052\u7b49\u6620\u5c04\u3002\u9ad8\u7ef4\u4e0d\u662f&#034;\u88c5\u4e0d\u8fdb\u4e09\u7ef4&#034;&#xff0c;\u800c\u662f\u6295\u5f71\u4e0b\u6765\u53ea\u5269\u4e09\u7ef4\u3002<\/p>\n<p>\u63a8\u8bba 8.3&#xff08;\u5ea6\u5236\u65e0\u5173\u6027&#xff09;\u3000\u672c\u539f\u7406\u7684\u5168\u90e8\u5224\u636e\u5747\u4e3a\u4e0d\u53d8\u91cf&#xff0c;\u4e0d\u4f9d\u8d56\u89d2\u5ea6\u5355\u4f4d\u7684\u9009\u53d6\u3002&#034;90\u00b0&#034;\u4ec5\u662f\u6b63\u4ea4\u5173\u7cfb\u5728 360 \u5ea6\u5236\u4e0b\u7684\u8bfb\u6570\u6807\u7b7e&#xff1b;\u6362\u7528\u767e\u5206\u5ea6\u3001\u5f27\u5ea6\u6216\u5468\u5929\u5f52\u4e00&#xff0c;\u539f\u7406\u4e00\u5b57\u4e0d\u6539\u3002<\/p>\n<p>\u63a8\u8bba 8.4&#xff08;\u6df7\u6c8c\u4e0e\u672c\u4f53\u7684\u7ed1\u5b9a&#xff09;\u3000\u6df7\u6c8c\u7684\u6839\u672c\u7279\u5f81\u662f\u7ea0\u7f20&#xff1b;\u7ea0\u7f20\u552f\u4e00\u5730\u7531\u4e09\u7ef4\u627f\u8f7d&#xff08;\u4e8b\u5b9e 6.1&#xff09;\u3002\u6545&#xff1a;\u6df7\u6c8c\u672c\u4f53\u82e5\u4f5c\u4e3a\u771f\u5b9e\u7ed3\u6784\u800c\u5b58\u5728&#xff0c;\u5176\u62d3\u6251\u821e\u53f0\u5fc5\u4e3a\u4e09\u7ef4&#xff0c;\u4e14\u4ec5\u4e3a\u4e09\u7ef4\u3002<\/p>\n<hr \/>\n<h3>9 \u7ed3\u8bba<\/h3>\n<li>\u7ef4\u5ea6\u7531\u4e24\u4e24\u6b63\u4ea4\u65b9\u5411\u7684\u6700\u5927\u6570\u76ee\u523b\u753b&#xff0c;\u6b63\u4ea4\u5373\u5185\u79ef\u4e3a\u96f6&#xff0c;\u4e0e\u89d2\u5ea6\u5355\u4f4d\u65e0\u5173&#xff08;\u00a72&#xff09;&#xff1b;<\/li>\n<li>\u7531\u6b63\u4ea4\u5bb9\u91cf\u5b9a\u7406&#xff0c;R\u00b3 \u4e2d\u4e24\u4e24\u6b63\u4ea4\u65b9\u5411\u81f3\u591a\u4e09\u4e2a&#xff08;\u00a73&#xff09;&#xff1b;<\/li>\n<li>\u800c R\u00b3 \u4e2d\u65b9\u5411\u6709\u65e0\u7a77\u591a\u2014\u2014\u6b63\u4ea4\u6027\u662f\u7a00\u7f3a\u8d44\u6e90&#xff0c;\u800c\u975e\u65b9\u5411\u7684\u7a00\u7f3a&#xff08;\u00a74&#xff09;&#xff1b;<\/li>\n<li>\u6545\u4efb\u4f55\u989d\u5916\u65b9\u5411\u5185\u5207\u4e8e\u4e09\u7ef4\u65f6&#xff0c;\u5fc5\u7136\u4e0e\u67d0\u4e00\u57fa\u65b9\u5411\u4e0d\u518d\u6b63\u4ea4&#xff08;\u4e09\u7ef4\u4e0b |u\u00b7e\u1d62| \u2265 1\/\u221a3&#xff09;&#xff0c;\u4ece\u800c\u9000\u5316\u4e3a\u7ebf\u6027\u7ec4\u5408&#xff0c;\u5305\u542b\u4e8e\u4e09\u7ef4&#xff08;\u00a74&#xff09;&#xff1b;<\/li>\n<li>\u4e00\u7ef4\u3001\u4e8c\u7ef4\u56e0\u6b63\u4ea4\u5173\u7cfb\u65e0\u4ece\u5efa\u7acb\u3001\u5185\u5916\u65e0\u6cd5\u754c\u5b9a\u800c\u9000\u5316&#xff08;\u00a75&#xff09;&#xff1b;<\/li>\n<li>\u7ebd\u7ed3\u9501\u5b9a\u3001Whitney \u95e8\u69db\u3001Poincar\u00e9 \u96be\u5ea6\u8c31\u3001\u53c9\u79ef\u95ed\u5408\u6027\u56db\u6761\u62d3\u6251\u5b66\u8bc1\u636e&#xff0c;\u4e0e Bertrand \u5b9a\u7406\u7684\u7269\u7406\u8bc1\u636e&#xff0c;\u4e00\u81f4\u5730\u786e\u8ba4\u4e09\u7ef4\u662f\u552f\u4e00\u7684\u4e34\u754c\u7ef4\u5ea6&#xff08;\u00a76\u20147&#xff09;\u3002<\/li>\n<p>\u4e00\u7ef4\u3001\u4e8c\u7ef4\u3001\u9ad8\u7ef4&#xff0c;\u5168\u90e8\u4e0d\u53ef\u6784\u9020\u3002\u4e09\u7ef4\u662f\u552f\u4e00\u7ef4\u5ea6\u3002<\/p>\n<p>\u8fd9\u4e00\u7ed3\u8bba\u4e0e\u6570\u5b66\u4e0a R\u00b9\u3001R\u00b2\u3001R\u2074\u2026 \u7684\u826f\u5b9a\u4e49\u6027\u5e76\u4e0d\u77db\u76fe\u2014\u2014\u4e8c\u8005\u5206\u5c5e\u4e0d\u540c\u5c42\u6b21&#xff1a;\u6570\u5b66\u662f\u63cf\u8ff0\u5de5\u5177&#xff0c;\u7269\u7406\u4e0e\u6784\u9020\u662f\u88ab\u63cf\u8ff0\u7684\u5bf9\u8c61\u3002\u5728\u6784\u9020\u5c42\u6b21\u4e0a&#xff0c;\u4e09\u7ef4\u552f\u4e00\u3002<\/p>\n<p>\u5bf9\u6df7\u6c8c\u62d3\u6251\u5b66\u800c\u8a00&#xff0c;\u8fd9\u4e00\u539f\u7406\u5177\u6709\u5960\u57fa\u610f\u4e49&#xff1a;\u6df7\u6c8c\u4f5c\u4e3a\u7ea0\u7f20\u7684\u672c\u4f53&#xff0c;\u5176\u53ef\u5b58\u5728\u7684\u62d3\u6251\u821e\u53f0\u662f\u552f\u4e00\u7684\u3002\u800c\u90a3\u552f\u4e00\u8005&#xff0c;\u6b63\u662f\u6211\u4eec\u8eab\u5904\u5176\u4e2d\u7684\u4e09\u7ef4\u3002<\/p>\n<hr \/>\n<h3>\u53c2\u8003\u6587\u732e<\/h3>\n<li>Brouwer, L. E. J. (1911). Beweis der Invarianz der Dimensionenzahl. Mathematische Annalen, 70, 161\u2013165.<\/li>\n<li>Bertrand, J. (1873). Comptes Rendus, 77, 849\u2013853.<\/li>\n<li>Ehrenfest, P. (1917). In what way does it become manifest in the fundamental laws of physics that space has three dimensions? Proc. Royal Netherlands Academy, 20, 200\u2013209.<\/li>\n<li>Whitney, H. (1944). The self-intersections of a smooth n-manifold in 2n-space. Annals of Mathematics, 45, 220\u2013246.<\/li>\n<li>Smale, S. (1961). Generalized Poincar\u00e9\u2019s conjecture in dimensions greater than four. Annals of Mathematics, 74, 391\u2013406.<\/li>\n<li>Freedman, M. H. (1982). The topology of four-dimensional manifolds. Journal of Differential Geometry, 17, 357\u2013453.<\/li>\n<li>Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math\/0211159.<\/li>\n<li>Perelman, G. (2003). Ricci flow with surgery on three-manifolds. arXiv:math\/0303109.<\/li>\n<li>Rolfsen, D. (1976). Knots and Links. Publish or Perish.<\/li>\n<li>Hurwitz, A. (1898). \u00dcber die Composition der quadratischen Formen. Nachr. Ges. Wiss. G\u00f6ttingen, 309\u2013316.<\/li>\n<li>Adams, J. F. (1960). On the non-existence of elements of Hopf invariant one. Annals of Mathematics, 72, 20\u2013104.<\/li>\n<li>Massey, W. S. (1967). Algebraic Topology: An Introduction.<\/li>\n<li>Conway, J. H., &amp; Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups (3rd ed.). Springer.<\/li>\n<li>Brouwer, L. E. J. (1911). Beweis der Invarianz der Dimensionenzahl. Mathematische Annalen, 70, 161\u2013165.<\/li>\n<li>Bertrand, J. (1873). Comptes Rendus, 77, 849\u2013853.<\/li>\n<li>Ehrenfest, P. (1917). In what way does it become manifest in the fundamental laws of physics that space has three dimensions? Proc. Royal Netherlands Academy, 20, 200\u2013209.<\/li>\n<li>Whitney, H. (1944). The self-intersections of a smooth n-manifold in 2n-space. Annals of Mathematics, 45, 220\u2013246.<\/li>\n<li>Smale, S. (1961). Generalized Poincar\u00e9\u2019s conjecture in dimensions greater than four. Annals of Mathematics, 74, 391\u2013406.<\/li>\n<li>Freedman, M. H. (1982). The topology of four-dimensional manifolds. Journal of Differential Geometry, 17, 357\u2013453.<\/li>\n<li>Perelman, G. (2002\/2003). The entropy formula for the Ricci flow; Ricci flow with surgery on three-manifolds. arXiv:math\/0211159, arXiv:math\/0303109.<\/li>\n<li>Rolfsen, D. (1976). Knots and Links. Publish or Perish.<\/li>\n<li>Hurwitz, A. (1898). \u00dcber die Composition der quadratischen Formen. Nachr. Ges. Wiss. G\u00f6ttingen, 309\u2013316.<\/li>\n<li>Adams, J. F. (1960). On the non-existence of elements of Hopf invariant one. Annals of Mathematics, 72, 20\u2013104.<\/li>\n<li>Massey, W. S. (1967). Algebraic Topology: An Introduction.<\/li>\n<li>Conway, J. H., &amp; Sloane, N. J. A. (1999). Sphere Packings, Lattices and Groups (3rd ed.). Springer.<\/li>\n<hr \/>\n<p>\u672c\u6587\u7684\u6838\u5fc3\u5224\u636e\u2014\u2014\u201c\u5185\u5207\u65f6\u4e0e\u65e2\u6709\u65b9\u5411\u4e0d\u518d\u6b63\u4ea4&#xff08;\u5728 360 \u5ea6\u5236\u4e0b\u8bfb\u4f5c&#xff1a;\u90a3\u4e2a\u89d2\u5df2\u4e0d\u518d\u662f 90 \u5ea6&#xff09;\u201d\u2014\u2014\u7531\u738b\u78ca\u63d0\u51fa&#xff1b;\u5176\u4e0d\u53d8\u91cf\u5f62\u5f0f&#xff08;\u5185\u79ef\u4e3a\u96f6&#xff09;\u53ca\u5ea6\u5236\u65e0\u5173\u6027\u8bba\u8bc1\u89c1 \u00a72.3\u20132.4&#xff0c;\u5b9a\u7406\u8bc1\u660e\u89c1 \u00a73\u20136\u3002\u672c\u6587\u7531\u738b\u78ca\u7684\u81ed\u5b9d\u5143\u5b9d\u521d\u5a9b\u8bba\u8bc1\u6392\u7248<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6458\u8981<br \/>\n\u672c\u6587\u5728\u6df7\u6c8c\u62d3\u6251\u5b66\u7684\u6846\u67b6\u4e0b&#xff0c;\u63d0\u51fa\u5e76\u4e25\u683c\u8bc1\u660e\u4e00\u6761\u57fa\u672c\u5b9a\u7406&#xff1a;\u4e09\u7ef4\u662f\u552f\u4e00\u7ef4\u5ea6\u3002\u7ef4\u5ea6\u552f\u4e00\u6027<br \/>\n\u8bba\u8bc1\u7684\u6838\u5fc3&#xff0c;\u662f\u628a\\&#8221;\u7ef4\u5ea6\\&#8221;\u7684\u5224\u636e\u4ece\\&#8221;\u65b9\u5411\u7684\u6761\u6570\\&#8221;\u6539\u4e3a\\&#8221;\u80fd\u5426\u4e0e\u65e2\u6709\u65b9\u5411\u6b63\u4ea4\\&#8221;\u3002\u5728\u6b64\u5224\u636e\u4e0b&#xff1a;<br \/>\n\u4e00\u7ef4&#xff1a;\u6b63\u4ea4\u662f\u4e8c\u5143\u5173\u7cfb&#xff0c;\u4e00\u7ef4\u4e2d\u65e0\u4ece\u5efa\u7acb&#xff0c;\u6545\u9000\u5316&#xff1b;\u4e8c\u7ef4&#xff1a;\u7f3a\u7b2c\u4e09\u6761\u4e92\u76f8\u6b63\u4ea4\u7684\u65b9\u5411&#xff0c;\u65e0\u6cd5\u754c\u5b9a\\&#8221;\u9762\u5185&#xff0f;\u9762\u5916\\&#8221;&#xff0c;\u6545\u4e0d\u53ef\u81ea\u8db3&#xff1b;\u4e09\u7ef4&amp;#x<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[12392,50,4508,427],"topic":[],"class_list":["post-116899","post","type-post","status-publish","format-standard","hentry","category-server","tag-12392","tag-50","tag-4508","tag-427"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/116899.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"og:description\" content=\"\u6458\u8981 \u672c\u6587\u5728\u6df7\u6c8c\u62d3\u6251\u5b66\u7684\u6846\u67b6\u4e0b&#xff0c;\u63d0\u51fa\u5e76\u4e25\u683c\u8bc1\u660e\u4e00\u6761\u57fa\u672c\u5b9a\u7406&#xff1a;\u4e09\u7ef4\u662f\u552f\u4e00\u7ef4\u5ea6\u3002\u7ef4\u5ea6\u552f\u4e00\u6027 \u8bba\u8bc1\u7684\u6838\u5fc3&#xff0c;\u662f\u628a&quot;\u7ef4\u5ea6&quot;\u7684\u5224\u636e\u4ece&quot;\u65b9\u5411\u7684\u6761\u6570&quot;\u6539\u4e3a&quot;\u80fd\u5426\u4e0e\u65e2\u6709\u65b9\u5411\u6b63\u4ea4&quot;\u3002\u5728\u6b64\u5224\u636e\u4e0b&#xff1a; \u4e00\u7ef4&#xff1a;\u6b63\u4ea4\u662f\u4e8c\u5143\u5173\u7cfb&#xff0c;\u4e00\u7ef4\u4e2d\u65e0\u4ece\u5efa\u7acb&#xff0c;\u6545\u9000\u5316&#xff1b;\u4e8c\u7ef4&#xff1a;\u7f3a\u7b2c\u4e09\u6761\u4e92\u76f8\u6b63\u4ea4\u7684\u65b9\u5411&#xff0c;\u65e0\u6cd5\u754c\u5b9a&quot;\u9762\u5185&#xff0f;\u9762\u5916&quot;&#xff0c;\u6545\u4e0d\u53ef\u81ea\u8db3&#xff1b;\u4e09\u7ef4&amp;#x\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.wsisp.com\/helps\/116899.html\" \/>\n<meta property=\"og:site_name\" content=\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\" \/>\n<meta property=\"article:published_time\" content=\"2026-10-22T00:24:43+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/116899.html\",\"url\":\"https:\/\/www.wsisp.com\/helps\/116899.html\",\"name\":\"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\",\"isPartOf\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/#website\"},\"datePublished\":\"2026-10-22T00:24:43+00:00\",\"dateModified\":\"2026-10-22T00:24:43+00:00\",\"author\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41\"},\"breadcrumb\":{\"@id\":\"https:\/\/www.wsisp.com\/helps\/116899.html#breadcrumb\"},\"inLanguage\":\"zh-Hans\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.wsisp.com\/helps\/116899.html\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/116899.html#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"\u9996\u9875\",\"item\":\"https:\/\/www.wsisp.com\/helps\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#website\",\"url\":\"https:\/\/www.wsisp.com\/helps\/\",\"name\":\"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3\",\"description\":\"\u9999\u6e2f\u670d\u52a1\u5668_\u9999\u6e2f\u4e91\u670d\u52a1\u5668\u8d44\u8baf_\u670d\u52a1\u5668\u5e2e\u52a9\u6587\u6863_\u670d\u52a1\u5668\u6559\u7a0b\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.wsisp.com\/helps\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"zh-Hans\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41\",\"name\":\"admin\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-Hans\",\"@id\":\"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery\",\"contentUrl\":\"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery\",\"caption\":\"admin\"},\"sameAs\":[\"http:\/\/wp.wsisp.com\"],\"url\":\"https:\/\/www.wsisp.com\/helps\/author\/admin\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.wsisp.com\/helps\/116899.html","og_locale":"zh_CN","og_type":"article","og_title":"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","og_description":"\u6458\u8981 \u672c\u6587\u5728\u6df7\u6c8c\u62d3\u6251\u5b66\u7684\u6846\u67b6\u4e0b&#xff0c;\u63d0\u51fa\u5e76\u4e25\u683c\u8bc1\u660e\u4e00\u6761\u57fa\u672c\u5b9a\u7406&#xff1a;\u4e09\u7ef4\u662f\u552f\u4e00\u7ef4\u5ea6\u3002\u7ef4\u5ea6\u552f\u4e00\u6027 \u8bba\u8bc1\u7684\u6838\u5fc3&#xff0c;\u662f\u628a\"\u7ef4\u5ea6\"\u7684\u5224\u636e\u4ece\"\u65b9\u5411\u7684\u6761\u6570\"\u6539\u4e3a\"\u80fd\u5426\u4e0e\u65e2\u6709\u65b9\u5411\u6b63\u4ea4\"\u3002\u5728\u6b64\u5224\u636e\u4e0b&#xff1a; \u4e00\u7ef4&#xff1a;\u6b63\u4ea4\u662f\u4e8c\u5143\u5173\u7cfb&#xff0c;\u4e00\u7ef4\u4e2d\u65e0\u4ece\u5efa\u7acb&#xff0c;\u6545\u9000\u5316&#xff1b;\u4e8c\u7ef4&#xff1a;\u7f3a\u7b2c\u4e09\u6761\u4e92\u76f8\u6b63\u4ea4\u7684\u65b9\u5411&#xff0c;\u65e0\u6cd5\u754c\u5b9a\"\u9762\u5185&#xff0f;\u9762\u5916\"&#xff0c;\u6545\u4e0d\u53ef\u81ea\u8db3&#xff1b;\u4e09\u7ef4&amp;#x","og_url":"https:\/\/www.wsisp.com\/helps\/116899.html","og_site_name":"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","article_published_time":"2026-10-22T00:24:43+00:00","author":"admin","twitter_card":"summary_large_image","twitter_misc":{"\u4f5c\u8005":"admin","\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4":"8 \u5206"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/www.wsisp.com\/helps\/116899.html","url":"https:\/\/www.wsisp.com\/helps\/116899.html","name":"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","isPartOf":{"@id":"https:\/\/www.wsisp.com\/helps\/#website"},"datePublished":"2026-10-22T00:24:43+00:00","dateModified":"2026-10-22T00:24:43+00:00","author":{"@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41"},"breadcrumb":{"@id":"https:\/\/www.wsisp.com\/helps\/116899.html#breadcrumb"},"inLanguage":"zh-Hans","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.wsisp.com\/helps\/116899.html"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.wsisp.com\/helps\/116899.html#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"\u9996\u9875","item":"https:\/\/www.wsisp.com\/helps"},{"@type":"ListItem","position":2,"name":"\u6df7\u6c8c\u62d3\u6251\u5b66(HDT)\u4e09\u7ef4\u552f\u4e00\u539f\u7406 \u5bf9\u5f26\u8bba\u9ad8\u7ef4\u5047\u8bf4\u7684\u5256\u6790"}]},{"@type":"WebSite","@id":"https:\/\/www.wsisp.com\/helps\/#website","url":"https:\/\/www.wsisp.com\/helps\/","name":"\u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3","description":"\u9999\u6e2f\u670d\u52a1\u5668_\u9999\u6e2f\u4e91\u670d\u52a1\u5668\u8d44\u8baf_\u670d\u52a1\u5668\u5e2e\u52a9\u6587\u6863_\u670d\u52a1\u5668\u6559\u7a0b","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.wsisp.com\/helps\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"zh-Hans"},{"@type":"Person","@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/358e386c577a3ab51c4493330a20ad41","name":"admin","image":{"@type":"ImageObject","inLanguage":"zh-Hans","@id":"https:\/\/www.wsisp.com\/helps\/#\/schema\/person\/image\/","url":"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery","contentUrl":"https:\/\/gravatar.wp-china-yes.net\/avatar\/?s=96&d=mystery","caption":"admin"},"sameAs":["http:\/\/wp.wsisp.com"],"url":"https:\/\/www.wsisp.com\/helps\/author\/admin"}]}},"_links":{"self":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts\/116899","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/comments?post=116899"}],"version-history":[{"count":0,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/posts\/116899\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/media?parent=116899"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/categories?post=116899"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/tags?post=116899"},{"taxonomy":"topic","embeddable":true,"href":"https:\/\/www.wsisp.com\/helps\/wp-json\/wp\/v2\/topic?post=116899"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}