Geodesic-preserving bijections of the Thurston geometries

Ryan Dickmann; Palani Lideros; Akash Narayanan

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arXiv · 2510.25585

Geodesic-preserving bijections of the Thurston geometries

Ryan Dickmann·Palani Lideros·Akash Narayanan

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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.

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Ryan Dickmann, Palani Lideros, Akash Narayanan. 2025-10-29. Geodesic-preserving bijections of the Thurston geometries. https://arxiv.org/abs/2510.25585

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