Geodesic-preserving bijections of the Thurston geometries

TL;DR

This paper classifies all bijections of Thurston geometries that preserve geodesics, providing a complete understanding of their structure and behavior across different geometrical settings.

Contribution

It offers a complete classification of geodesic-preserving bijections in Thurston geometries and related surfaces, extending understanding of geometric symmetries.

Findings

Bijections of Thurston geometries that preserve geodesics are fully classified.

In certain Riemannian manifolds, totally geodesic subsets are shown to be submanifolds.

Specific classifications are provided for the Euclidean cylinder and hyperbolic plane.

Abstract

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves.