{"id":355,"date":"2013-06-28T21:38:42","date_gmt":"2013-06-28T13:38:42","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=355"},"modified":"2013-06-28T21:38:42","modified_gmt":"2013-06-28T13:38:42","slug":"hilbert%e7%ac%ac%e5%9b%9b%e9%97%ae%e9%a2%98%e4%b8%8e%e5%b0%84%e5%bd%b1%e5%b9%b3%e5%9d%a6%e6%b5%81%e5%bd%a2%e7%9a%84%e5%88%86%e7%b1%bb","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=355","title":{"rendered":"Hilbert\u7b2c\u56db\u95ee\u9898\u4e0e\u5c04\u5f71\u5e73\u5766\u6d41\u5f62\u7684\u5206\u7c7b"},"content":{"rendered":"<h4 id=\"hilbert1\">Hilbert\u5ea6\u91cf[<a href=\"http:\/\/www.cms.zju.edu.cn\/conference\/2007\/geometric\/Proj_metric_Li_Talk.pdf\">1<\/a>]<\/h4>\n<p>\u5728$\\R^n$\u4e2d\u4e00\u51f8\u533a\u57df$\\Omega$\u4e0a, Hilbert\u5b9a\u4e49\u4e86\u6240\u8c13\u7684<strong>Hilbert\u5ea6\u91cf<\/strong>:<br \/>\n\\[<br \/>\nd_\\Omega(x,y)=\\frac{1}{2}\\log[a,x,y,b]=\\frac{1}{2}\\log\\frac{|y-a||x-b|}{|x-a||y-b|},\\quad x,y\\in\\Omega,\\: a,b\\in\\pt\\Omega.<br \/>\n\\]<br \/>\n\u7279\u522b\u5730,<br \/>\n\\[<br \/>\n(\\Omega,d_\\Omega)=\\begin{cases}<br \/>\n\\text{Minkowski geometry},&amp;\\Omega \u4e2d\u5fc3\u5bf9\u79f0\\\\<br \/>\n\\text{Lobachevskii geometry},&amp;\\Omega\u662f\u692d\u7403\\\\<br \/>\n\\text{hyperbolic geometry(Klein\u6a21\u578b)},&amp;\\Omega\u662f\u5355\u4f4d\u7403B^n(1).<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n<!--more--><\/p>\n<h4 id=\"hilbert\">Hilbert\u7b2c\u56db\u95ee\u9898<\/h4>\n<p>Hilbert\u7b2c\u56db\u95ee\u9898\u662f\u8bf4:<br \/>\n\\begin{prob}<br \/>\n\u6c42\u51fa$\\Omega$\u4e0a\u6240\u6709\u4f7f\u5f97\u6d4b\u5730\u7ebf\u662f\u76f4\u7ebf\u7684\u5ea6\u91cf. \u8fd9\u6837\u7684\u5ea6\u91cf\u4e5f\u79f0\u4e3a\u662f<strong>\u5c04\u5f71\u5e73\u5766<\/strong>\u7684.<br \/>\n\\end{prob}<br \/>\n\u4e00\u4e2a\u5149\u6ed1\u7684\u5ea6\u91cf\u79f0\u4e3a<strong>Finsler\u5ea6\u91cf<\/strong>, \u56e0\u6b64Hilbert\u7b2c\u56db\u95ee\u9898\u7684\u6b63\u5219\u89e3\u5c31\u662f\u5bfb\u627e(\u5c40\u90e8)\u5c04\u5f71\u5e73\u5766\u7684Finsler\u5ea6\u91cf. <strong>\u7279\u522bHilbert\u5ea6\u91cf\u662f\u5c04\u5f71\u5e73\u5766\u7684Finsler\u5ea6\u91cf<\/strong>.<\/p>\n<h4 id=\"\">\u5c40\u90e8\u5c04\u5f71\u5e73\u5766\u6d41\u5f62\u7684\u5206\u7c7b<\/h4>\n<p>\u56de\u5fc6\u5728\u9ece\u66fc\u51e0\u4f55\u4e2d, \u6211\u4eec\u6709<br \/>\n\\begin{thm}[Beltrami]<br \/>\n\u4e00\u4e2a\u9ece\u66fc\u6d41\u5f62\u662f\u5c40\u90e8\u5c04\u5f71\u5e73\u5766\u7684\u5f53\u4e14\u4ec5\u5f53\u5b83\u662f\u5e38\u622a\u9762\u66f2\u7387\u7684.<br \/>\n\\end{thm}<br \/>\n\u4f46\u662f\u8fd9\u5bf9\u4e00\u822c\u7684Finsler\u5ea6\u91cf\u4e0d\u518d\u6210\u7acb. \u5bf9\u4e00\u822c\u7684\u5c04\u5f71\u5e73\u5766\u7684Finsler\u5ea6\u91cf\u8fdb\u884c\u5206\u7c7b\u8fd8\u975e\u5e38\u9065\u8fdc, \u76ee\u524d\u6211\u4eec\u4e00\u822c\u90fd\u53ea\u5728\u5047\u8bbe\u5177\u6709\u5e38\u65d7\u66f2\u7387(\u5c04\u5f71\u5e73\u5766\u4e00\u5b9a\u5177\u6709\u6807\u91cf\u65d7\u66f2\u7387, \u8bb0\u4e0d\u4f9d\u8d56\u4e8e\u65d7\u6746, \u53ea\u4e0e\u70b9\u548c\u65b9\u5411\u6709\u5173)\u7684\u6761\u4ef6\u4e0b\u8ba8\u8bba. Z.Shen\u5bf9(\u975e\u6b63)\u5e38\u65d7\u66f2\u7387\u7684Randers\u5ea6\u91cf\u8fdb\u884c\u4e86\u5206\u7c7b:<br \/>\n\\begin{thm}[Shen<a id=\"fnref:shen\" class=\"footnote\" title=\"See footnote\" href=\"#fn:shen\">1<\/a>, 2003]<br \/>\n\u5047\u8bbe$F=\\alpha+\\beta$\u662f\u4e00\u4e2aRanders\u5ea6\u91cf, \u82e5\u5b83\u662f\u5c40\u90e8\u5c04\u5f71\u5e73\u5766\u7684\u4e14\u5177\u6709\u5e38\u65d7\u66f2\u7387$K\\leq0$, \u5219\u5f53$K=0$\u65f6, $F$\u662f\u5c40\u90e8Minkowski\u5ea6\u91cf, \u5f53$K&lt;0$\u65f6, $F$\u5c40\u90e8\u7b49\u8ddd\u4e8e\u5e7f\u4e49Funk\u5ea6\u91cf.<br \/>\n\\end{thm}<br \/>\n\u6211\u4eec\u77e5\u9053, \u6bd4Randers\u5ea6\u91cf\u68a2\u5e7f\u4e00\u70b9\u7684\u662f$(\\alpha,\\beta)$\u5ea6\u91cf. \u5173\u4e8e\u5c04\u5f71\u5e73\u5766\u7684$(\\alpha,\\beta)$\u5ea6\u91cf\u7684\u5206\u7c7b, \u6211\u4eec\u6709<br \/>\n\\begin{thm}[Li-shen<a id=\"fnref:lishen\" class=\"footnote\" title=\"See footnote\" href=\"#fn:lishen\">2<\/a>, 2007]<br \/>\n\u5b9a\u4e49\u5728$\\mathcal{U}\\subset\\R^n$, $n\\geq3$, \u4e0a\u7684$(\\alpha,\\beta)$\u5ea6\u91cf$F$\u662f\u5c04\u5f71\u5e73\u5766\u4e14\u5177\u6709\u5e38\u65d7\u66f2\u7387\u7684\u53ea\u6709\u5982\u4e0b\u4e09\u7c7b:<\/p>\n<ul>\n<li>$\\alpha$\u662f\u5c04\u5f71\u5e73\u5766\u7684\u9ece\u66fc\u5ea6\u91cf, \u800c$\\beta$\u5173\u4e8e$\\alpha$\u5e73\u884c;<\/li>\n<li>\u5f53\u5e38\u65d7\u66f2\u7387$K&lt;0$\u65f6, $F=\\sqrt{\\alpha^2+k\\beta^2}+\\eps\\beta$, \u8fd9\u91cc$\\eps\\neq0$, $k$\u90fd\u662f\u5e38\u6570;<\/li>\n<li>\u5f53\u5e38\u65d7\u66f2\u7387$K=0$\u65f6, $F=\\frac{\\left(\\sqrt{\\alpha^2+k\\beta^2}+\\eps\\beta\\right)^2}{\\sqrt{\\alpha^2+k\\beta^2}}$, \u8fd9\u91cc$\\eps\\neq0$, $k$\u90fd\u662f\u5e38\u6570.<\/li>\n<\/ul>\n<p>\\end{thm}<br \/>\n\u6b64\u5916, R. Bryant<a id=\"fnref:bry\" class=\"footnote\" title=\"See footnote\" href=\"#fn:bry\">3<\/a>\u5bf9$S^n$\u4e0a\u5177\u6709\u5e38\u65d7\u66f2\u7387$K=1$\u7684\u5c04\u5f71\u5e73\u5766Finsler\u5ea6\u91cf\u8fdb\u884c\u4e86\u5206\u7c7b.<br \/>\n\u6700\u8fd1, \u5bf9\u5177\u6709\u826f\u597d\u5bf9\u79f0\u6027\u7684Finsler\u5ea6\u91cf, Mo-Huang\u4e5f\u5f97\u5230\u4e00\u4e9b\u8fd9\u65b9\u9762\u7684\u7ed3\u679c.<br \/>\n\\begin{defn}[\u7403\u5bf9\u79f0\u5ea6\u91cf]\u5047\u8bbe$F$\u662f$B^n(r)$\u4e0a\u7684\u4e00\u4e2aFinsler\u5ea6\u91cf, \u5982\u679c<br \/>\n\\[<br \/>\nF(Ax,Ay)=F(x,y),\\quad\\forall x\\in B^n(r),\\quad \\forall y\\in T_xB^n(r),\\quad \\forall A\\in O(n),<br \/>\n\\]<br \/>\n\u90a3\u4e48\u6211\u4eec\u79f0$F$\u662f\u4e00\u4e2a<strong>\u7403\u5bf9\u79f0\u5ea6\u91cf<\/strong>.<br \/>\n\\end{defn}<br \/>\n\u5bb9\u6613\u770b\u51fa, \u5bf9\u79f0\u6027\u5c06Finsler\u5ea6\u91cf\u7684&#8221;\u590d\u6742\u5ea6&#8221;\u964d\u4f4e\u4e86, \u4e8b\u5b9e\u4e0a, Huang-Mo\u57282013\u7ed9\u51fa\u4e86\u7403\u5bf9\u79f0\u5ea6\u91cf\u7684\u4e00\u4e2a\u523b\u753b:<br \/>\n\\begin{lem}<br \/>\n$F$\u662f$B^n(r)$\u4e0a\u4e00\u4e2a\u7403\u5bf9\u79f0\u5ea6\u91cf\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u51fd\u6570$\\phi:[0,r)\\times \\R\\to\\R$\u4f7f\u5f97<br \/>\n\\[<br \/>\nF(x,y)=|y|\\phi\\left(|x|,\\langle x,y\\rangle\/|y|\\right),\\quad x,y\\in TB^n(r)\\setminus\\set{0}.<br \/>\n\\]<br \/>\n\u7531\u6b64\u53ef\u89c1\u6240\u6709\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf\u90fd\u662f\u5e7f\u4e49$(\\alpha,\\beta)$\u5ea6\u91cf,i.e., $F=\\alpha\\phi(\\|\\beta\\|_\\alpha,\\beta\/\\alpha)$.<br \/>\n\\end{lem}<br \/>\nHamel<a id=\"fnref:hamel\" class=\"footnote\" title=\"See footnote\" href=\"#fn:hamel\">4<\/a>\u57281903\u5e74\u7ed9\u51fa\u4e86$\\R^n$\u4e2d\u5c40\u90e8\u51f8\u533a\u57df\u4e0a\u5c04\u5f71\u5e73\u5766\u7684Finsler\u5ea6\u91cf\u7684\u523b\u753b:<br \/>\n\\begin{lem}[Hamel]<br \/>\n\u5047\u8bbe$\\Omega\\subset\\R^n$\u662f\u4e00\u4e2a\u51f8\u533a\u57df, \u5219$\\Omega$\u4e0a\u7684Finsler\u5ea6\u91cf$F$\u662f\u5c04\u5f71\u5e73\u5766\u7684\u5f53\u4e14\u4ec5\u5f53<br \/>\n\\[<br \/>\nF_{x^jy^i}y^j=F_{x^i},<br \/>\n\\]<br \/>\n\u8fd9\u4e5f\u7b49\u4ef7\u4e8e<br \/>\n\\[<br \/>\n\\frac{\\pt}{\\pt y^i}\\left(\\frac{\\pt{F}}{\\pt x^j}y^j\\right)=\\frac{\\pt F}{\\pt x^i}.<br \/>\n\\]<br \/>\n\\end{lem}<br \/>\n\u5229\u7528\u8fd9\u4e00\u523b\u753b, \u6211\u4eec\u53ef\u5f97\u5230\u7403\u5bf9\u79f0\u5ea6\u91cf\u5b9e\u5c04\u5f71\u5e73\u5766\u7684\u65b9\u7a0b:<br \/>\n\\begin{lem}<br \/>\n\u5047\u8bbe$F=|y|\\phi\\left(|x|,\\frac{\\langle x,y\\rangle}{|x||y|}\\right)$\u662f$B^n(r)$\u4e0a\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf, \u5219$F$\u662f\u5c04\u5f71\u5e73\u5766\u7684\u5f53\u4e14\u4ec5\u5f53<br \/>\n\\[<br \/>\ns\\phi_{bs}+b\\phi_{ss}-\\phi_s=0.<br \/>\n\\]<br \/>\n\\end{lem}<br \/>\n\u5bf9\u975e\u8d1f\u5e38\u65d7\u66f2\u7387\u4e14\u5c04\u5f71\u5e73\u5766\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf\u7684\u5206\u7c7b\u662f\u7531L. Zhou<a id=\"fnref:zhou\" class=\"footnote\" title=\"See footnote\" href=\"#fn:zhou\">5<\/a>\u5b8c\u6210\u7684:<br \/>\n\\begin{thm}<br \/>\n\u5047\u8bbe$F$\u662f$B^n(r)$\u4e0a\u4e00\u5177\u6709\u975e\u8d1f\u5e38\u65d7\u66f2\u7387$K$\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf, \u82e5\u4ed6\u8fd8\u662f\u5c04\u5f71\u5e73\u5766\u7684, \u5219\u5b83\u53ea\u80fd\u662f\u5982\u4e0b\u4e24\u7c7b:<\/p>\n<ul>\n<li>\u5f53 $K&gt;0$\u65f6, $F$\u662fBryant\u5ea6\u91cf, \u4e14\u5176\u53c2\u6570\u6ee1\u8db3$p_1=p_2=\\cdots=p_{n+1}$;<\/li>\n<li>\u5f53$K=0$\u662f, $F$\u662fBerwald\u5ea6\u91cf.<\/li>\n<\/ul>\n<p>\\end{thm}<br \/>\n\u800c\u5bf9\u8d1f\u5e38\u65d7\u66f2\u7387\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf, \u5728\u5c04\u5f71\u5e73\u5766\u65f6, Mo\u5f97\u5230\u5982\u4e0b\u7ed3\u679c:<br \/>\n\\begin{thm}<br \/>\n\u5047\u8bbe$F$\u662f$B^n(r)$\u4e0a\u4e00\u5177\u6709\u8d1f\u5e38\u65d7\u66f2\u7387$K=-1$\u7684\u7403\u5bf9\u79f0\u5ea6\u91cf, \u5219$F$\u662f\u5c04\u5f71\u5e73\u5766\u7684, \u5f53\u4e14\u4ec5\u5f53<br \/>\n\\[<br \/>\nF=\\frac{1}{2}\\left(\\theta_c(x,y)-\\eps\\theta_c(\\eps x,y)\\right),\\quad\\eps&lt;1,<br \/>\n\\]<br \/>\n\u5176\u4e2d$\\theta_c$\u662f\u4e25\u683c\u51f8\u57df$B^n(\\sqrt{c})$\u4e0a\u7684Funk\u5ea6\u91cf.<br \/>\n\\end{thm}<\/p>\n<hr \/>\n<blockquote><p>Written with <a href=\"http:\/\/benweet.github.io\/stackedit\/\">StackEdit<\/a>.<\/p><\/blockquote>\n<div class=\"footnotes\">\n<hr \/>\n<ol>\n<li id=\"fn:shen\">Shen, Zhongmin. &#8220;Projectively flat Finsler metrics of constant flag curvature.&#8221; <em>Transactions of the American Mathematical Society<\/em> 355.4 (2003): 1713-1728. <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:shen\">\u21a9<\/a><\/li>\n<li id=\"fn:lishen\">Li, Benling, and Zhongmin Shen. &#8220;On a class of projectively flat Finsler metrics with constant flag curvature.&#8221; International Journal of Mathematics 18.07 (2007): 749-760. <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:lishen\">\u21a9<\/a><\/li>\n<li id=\"fn:bry\">R. Bryant, <em>Some remarks on Finsler manifolds with constant flag curvature<\/em>, Houston J. of Math. 28(2002), 221-262. <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:bry\">\u21a9<\/a><\/li>\n<li id=\"fn:hamel\">G. Hamel,\u00dcber die Geometrieen in denen die Geraden die K\u00fcrzestensind, <em>Math. Ann<\/em>. 57(1903), 231-264. <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:hamel\">\u21a9<\/a><\/li>\n<li id=\"fn:zhou\">Zhou, Linfeng. &#8220;Projective spherically symmetric Finsler metrics with constant flag curvature in R n.&#8221; Geometriae Dedicata 158.1 (2012): 353-364. <a class=\"reversefootnote\" title=\"Return to article\" href=\"#fnref:zhou\">\u21a9<\/a><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hilbert\u5ea6\u91cf[1] \u5728$\\R^n$\u4e2d\u4e00\u51f8\u533a\u57df$\\Omega$\u4e0a, Hilb &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=355\"> <span class=\"screen-reader-text\">Hilbert\u7b2c\u56db\u95ee\u9898\u4e0e\u5c04\u5f71\u5e73\u5766\u6d41\u5f62\u7684\u5206\u7c7b<\/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":[475,507,196,115],"class_list":["post-355","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-finsler","tag-hilbert","tag-196","tag-115"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/355","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=355"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/355\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=355"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=355"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}