Geodesic-preserving bijections of the Thurston geometries

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.