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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