{"id":216,"date":"2013-06-19T22:22:38","date_gmt":"2013-06-19T14:22:38","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=216"},"modified":"2013-06-19T22:22:38","modified_gmt":"2013-06-19T14:22:38","slug":"2%e7%bb%b4randers%e5%ba%a6%e9%87%8f%e7%9a%84%e5%bc%a0%e9%87%8f%e5%88%bb%e7%94%bb","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=216","title":{"rendered":"2\u7ef4Randers\u5ea6\u91cf\u7684\u5f20\u91cf\u523b\u753b"},"content":{"rendered":"

\u5173\u4e8eFinsler\u51e0\u4f55\u4e00\u4e2a\u719f\u77e5\u7684\u7ed3\u679c\u662fMatsumoto\u53ca\u5176\u5b66\u751fH\u014dj\u014d\u7ed9\u51fa\u7684\u5728$n\\geq3$\u7684Finsler\u6d41\u5f62\u4e0a\u5ea6\u91cf\u662fRander’s\u5ea6\u91cf\u7684\u523b\u753b:
\n\\begin{thm}[M. Matsumoto & S. H\u014dj\u014d,1978]Let $F$ be a Minkowski norm on a vector space $V$ of dimension $n \\geq3$. The Matsumoto torsion $M= 0$ if and only if $F$ is a Randers norm.
\n\\end{thm}
\n\u6700\u8fd1, L, Mo and L-B, Huang(c.f.Mo&Huang2010<\/a>)\u5f97\u5230\u4e00\u4e2a\u523b\u753b2\u7ef4Finsler\u66f2\u9762\u4e0a\u5176Finsler\u5ea6\u91cf\u4e3aRander’s\u5ea6\u91cf\u7684\u5f20\u91cf$\\chi$.
\n\\begin{thm}[Mo&Huang, 2010]
\n Let $F$ be a Minkowski norm on a plane $V$. Then $F$ is a Randers norm if and only if $\u03c7$ is constant along the indicatrix<\/strong>.
\n\\end{thm}
\n\u5176\u65b9\u6cd5\u662f\u5229\u7528\u4eff\u5c04\u5fae\u5206\u51e0\u4f55\u4e0eFinsler\u51e0\u4f55\u7684\u5bf9\u5e94: Let $\\iota$ be the transversal field and $\\nabla$ be the induced affine connection of $\\iota$. Set $[y]_+=\\set{\\lambda y|\\lambda>0}$, and $\\mathbb{S}=\\set{[y]_+|y\\in V\\setminus{0}}$, the immersion induced by $F$ is defined by
\n\\begin{align*}
\nS_F:& \\mathbb{S} \\to V\\\\
\n[y]_+&\\mapsto\\frac{y}{F(y)}.
\n\\end{align*}<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Finsler\u51e0\u4f55<\/th>\n\u4eff\u5c04\u5fae\u5206\u51e0\u4f55<\/th>\n<\/tr>\n<\/thead>\n
angular metric $h$<\/td>\naffine fundamental form $h$<\/td>\n<\/tr>\n
<\/td>\n$\\nabla h$ fully symmetric cubic form induced by $\\iota$<\/td>\n<\/tr>\n
Cartan tensor $\\mathbb{C}$<\/td>\n$\\mathbb{C}=\\frac{1}{2}\\nabla h$<\/td>\n<\/tr>\n
main scalar $I$<\/td>\n<\/td>\n<\/tr>\n
$\\tau$ distortion of $F$<\/td>\n$\\phi=\\exp(\\frac{2\\tau}{3})$<\/td>\n<\/tr>\n
<\/td>\n$\\mathbb{Z}=-(\\rd\\phi)^\\sharp$<\/td>\n<\/tr>\n
<\/td>\n$\\tilde\\iota=\\phi\\iota+S_*^F(\\mathbb{Z})$<\/td>\n<\/tr>\n
<\/td>\naffine shape operator $\\tilde{s}$of $\\tilde\\iota$<\/td>\n<\/tr>\n
$\\chi=e^{\\frac{2\\tau}{3}}\\left(1+\\frac{2}{3}I’-\\frac{2}{9}I^2\\right)$<\/td>\nmean affine curvature $\\chi=\\mathrm{trace}{(\\tilde s)}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Thus, $\\chi$ is a constant iff $\\mathrm{trace}(\\tilde s)$ is constant iff $S^F$ of $F$ is an ellipsoid(c.f. Nomizu&Sasaki1994<\/a>) iff F is a Rander’s metric.
\n\\begin{prob}
\nQ: What’s the relationship with $M$ and $\\chi$? ($\\chi$ has higher dimensional generalization!)
\n\\end{prob}<\/p>\n

Reference<\/h4>\n