Finiteness of totally geodesic hypersurfaces

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.

arXiv comment30 pages, v2. minor revisions, v3. comments on smoothness vs analyticity added