2维Randers度量的张量刻画

关于Finsler几何一个熟知的结果是Matsumoto及其学生Hōjō给出的在$n\geq3$的Finsler流形上度量是Rander’s度量的刻画:

定理 1 (M. Matsumoto & S. Hōjō,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.

最近, L, Mo and L-B, Huang(c.f.Mo&Huang2010)得到一个刻画2维Finsler曲面上其Finsler度量为Rander’s度量的张量$\chi$.
定理 2 (Mo&Huang, 2010). Let $F$ be a Minkowski norm on a plane $V$. Then $F$ is a Randers norm if and only if $χ$ is constant along the indicatrix.

其方法是利用仿射微分几何与Finsler几何的对应: 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
\begin{align*}
S_F:& \mathbb{S} \to V\\
[y]_+&\mapsto\frac{y}{F(y)}.
\end{align*}

Finsler几何 仿射微分几何
angular metric $h$ affine fundamental form $h$
$\nabla h$ fully symmetric cubic form induced by $\iota$
Cartan tensor $\mathbb{C}$ $\mathbb{C}=\frac{1}{2}\nabla h$
main scalar $I$
$\tau$ distortion of $F$ $\phi=\exp(\frac{2\tau}{3})$
$\mathbb{Z}=-(\rd\phi)^\sharp$
$\tilde\iota=\phi\iota+S_*^F(\mathbb{Z})$
affine shape operator $\tilde{s}$of $\tilde\iota$
$\chi=e^{\frac{2\tau}{3}}\left(1+\frac{2}{3}I’-\frac{2}{9}I^2\right)$ mean affine curvature $\chi=\mathrm{trace}{(\tilde s)}$

Thus, $\chi$ is a constant iff $\mathrm{trace}(\tilde s)$ is constant iff $S^F$ of $F$ is an ellipsoid(c.f. Nomizu&Sasaki1994) iff F is a Rander’s metric.

问题 1. Q: What’s the relationship with $M$ and $\chi$? ($\chi$ has higher dimensional generalization!)

Reference

  • [Mo&Huang2010] MO, XIAOHUAN, and LIBING HUANG. “On characterizations of Randers norms in a Minkowski space.” International Journal of Mathematics 21.04 (2010): 523-535.
  • [Nomizu&Sasaki1994] Nomizu, Katsumi, and Takeshi Sasaki. Affine differential geometry: geometry of affine immersions. Cambridge University Press, 1994.

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