论文标题
严格凸组和闭合凸的射凸表面上的射击，保留完整的大地测量学
Bijections on strictly convex sets and closed convex projective surfaces that preserve complete geodesics
论文作者
论文摘要
在本文中，我们研究了$ \ Mathbf r \ Mathbf P^n $的严格凸组集合，$ n \ geq 2 $和闭合的凸影投影表面，配备了希尔伯特度量标准，可将其映射到完整的大地测量学以完成地理位置。双曲线$ n $空间及其标准度量是我们考虑的空间的一个特殊示例，众所周知，在这种情况下，这些徒恰好是异构体。我们首先证明该结果概括为一个任意的严格凸集。对于表面设置，我们证明了将简单的封闭测量学映射到简单封闭的大地测量学和映射封闭的大地测量学的等效性。我们还概述了一些未来的方向和问题，以进一步探讨这些主题。
In this paper, we study bijections on strictly convex sets of $\mathbf R \mathbf P^n$ for $n \geq 2$ and closed convex projective surfaces equipped with the Hilbert metric that map complete geodesics to complete geodesics as sets. Hyperbolic $n$-space with its standard metric is a special example of the spaces we consider, and it is known that these bijections in this context are precisely the isometries. We first prove that this result generalizes to an arbitrary strictly convex set. For the surfaces setting, we prove the equivalence of mapping simple closed geodesics to simple closed geodesics and mapping closed geodesics to closed geodesics. We also outline some future directions and questions to further explore these topics.
