Finiteness of totally geodesic hypersurfaces
Differential Geometry
2025-11-17 v3 Dynamical Systems
Geometric Topology
Abstract
We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold with negative sectional curvature has only finitely many totally geodesic hypersurfaces, unless it has constant curvature.
Cite
@article{arxiv.2408.03430,
title = {Finiteness of totally geodesic hypersurfaces},
author = {Simion Filip and David Fisher and Ben Lowe},
journal= {arXiv preprint arXiv:2408.03430},
year = {2025}
}
Comments
30 pages, v2. minor revisions, v3. comments on smoothness vs analyticity added