常曲率Randers度量的分类

我们知道, 在黎曼几何中, 关于常截面曲率的完备黎曼流形(即空间形式)的分类已经完全解决:

定理 1. 对每个$c\in\R$以及所有的$n\in\Z^+$, 都存在唯一的(只相差一个等距)的单连通的$n$为空间形式, 使得其常截面曲率为$c$.

证明 . c.f. [伍鸿熙1989] P70 Thm1 与 P97 Thm10.

2004年, D, Bao, C, Robles & Z, Shen 证明了关于常旗曲率的Randers度量的分类定理, 证明主要依赖于如下结果:

定理 2. Let $F$ be a Randers metric and $(h,v)$ be its navigation representation. Then $F$ has constant flag curvature iff $(h,v)$ satisfies:
$$
\begin{cases}
v_{i|j}+v_{j|i}=-4c h_{ij}\tag{1}\\
K=\mu-c^2
\end{cases}
$$
where $\mu$ is the constant sectional curvature of the Riemannian metric $h$, $K$ is the constant flag curvature of $F$, and $c$ is a constant.

note that the first equatiuon of \eqref{eq:1} implies that $v$ is homothetic with respect to $h$, i.e., $L_vg=-2c g$, thus, we can solve it for any given $c$ (in the case that $h$ is of constant sectional curvature $\mu$).

Given a Finsler metric $F$ and a vector field $v$ with $F(x,v_x)<1$, we can define a new Finsler metric $\tilde F$ by
\begin{equation}
F(x,y/\tilde F(x,y)+v_x)=1.
\end{equation}

定理 3 (Chern-Shen, Lemma 1.4.1). For any Piecewise $C^\infty$ curve $c$ in $M$, the $\tilde F$-length of $c$ is equal to the time for which the object travels along $c$.

特别, 当$F$是一个黎曼度量时, 可以证明导航问题的解 $\tilde F$ 是一个Randers度量, 而且反过来也是对的. 即, 给定一个Randers度量, 它可以看出在某个黎曼度量下导航问题的解. 具体的证明可以参考我的笔记:Notes on Riemannian-Finsler Geometry (Prop 3.1).

后来, Robles 在2007年得到了常旗曲率的Randers度量的所有测地线.

定义 4. A smooth curve in a Finsler manifold is called a geodesic if it is locally the shortest path connecting two points on this curve.

定理 5 (C. Robles, 2007). Let $F$ be a Randers metric of constant flag curvature and $(h,v)$ be its navigational representation, then the geodesic of $F$ are given by $\psi_t(\gamma(a(t)))$, where $\psi_t$ is the flow of $-v$ and $\gamma(t)$ is a geodesic of $h$ and $a(t)$ is defined by
$$
a(t)=\begin{cases}
\frac{e^{2ct}-1}{2c},&c\neq0\
t,&c=0.
\end{cases}
$$

References

  1. 伍鸿照, 沈纯理, and 虞言林. “黎曼几何初步.” 北京大学出版社,(1989).
  2. Bao, David, Colleen Robles, and Zhongmin Shen. “Zermelo navigation on Riemannian manifolds.” Journal of Differential Geometry 66.3 (2004): 377-435.
  3. http://www.docin.com/p-51194347.html
  4. Robles, Colleen. “Geodesics in Randers spaces of constant curvature.” Transactions of the American Mathematical Society (2007): 1633-1651.

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