Bijections on strictly convex sets and closed convex projective surfaces that preserve complete geodesics

TL;DR

This paper investigates bijections on strictly convex sets and convex projective surfaces that preserve complete geodesics, generalizing known isometry results and exploring geodesic mappings.

Contribution

It extends the characterization of geodesic-preserving bijections from hyperbolic space to general strictly convex sets and analyzes their properties on convex projective surfaces.

Findings

Bijections on strictly convex sets that preserve complete geodesics are characterized as isometries.

Mapping simple closed geodesics to simple closed geodesics is equivalent to mapping all closed geodesics.

The results generalize hyperbolic space isometry to broader convex geometric contexts.

Abstract

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.