常曲率空间的Funk几何和Hilbert几何
The Funk and Hilbert geometries for spaces of constant curvature
摘要：
本文的目的是介绍和研究双曲或球面空间的开凸子集$Ω$上的Euclidean Funk和Hilbert度量的类似物。至少在形式层面上，这三种情况有着惊人的相似之处：欧几里得、球面和双曲。我们首先定义非欧几里德类似物的欧几里德芬克弱度量，我们给三个不同的表示，它在每个非欧几里德的情况下，平行的已知情况下的欧几里德的情况。因此，所有这些指标被证明是Finsler的，并描述了相关规范的Finsler度量。利用球面$S^n$和双曲空间$\mathbb{H}^n$上的一组经典三角恒等式以及常曲率的非欧几里得空间上的交比定义，发展了该理论.这反过来又导致了在这些空间中的投射不变性的概念。然后，我们研究了测地线的芬克和希尔伯特度量。在欧几里得几何（分别为球面几何和双曲几何）的情况下，欧几里得（分别为球面几何和双曲几何）测地线是芬克测地线和希尔伯特测地线。在$\mathbb{R}^n$、$\mathbb{H}^n$空间和上半球之间存在的自然投影映射证明了Hilbert凸集几何在这三个常曲率空间中的理论都是等价的。但对于Funk几何体来说，情况却并非如此。
英文摘要：
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets $Ω$ of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representations of it in each of the non-Euclidean cases, which parallel the known situation for the Euclidean case. As a consequence, all of these metrics are shown to be Finslerian, and the associated norms of the Finsler metrics are described. The theory is developed by using a set of classical trigonometric identities on the sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$ and the definition of a cross ratio on the non-Euclidean spaces of constant curvature. This in turn leads to the concept of projectivity invariance in these spaces. We then study the geodesics of the Funk and Hilbert metrics. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. Natural projection maps that exist between the spaces $\mathbb{R}^n$, $\mathbb{H}^n$ and the upper hemisphere demonstrate that the theories of Hilbert geometry of convex sets in the three spaces of constant curvature are all equivalent. The same cannot be said about the Funk geometries.
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