I will explain the proof, joint with David Fisher and Ben Lowe, that any closed analytic Riemannian manifold with negative sectional curvature has only finitely many totally geodesic hypersurfaces, unless it has constant curvature. When such a constant negative curvature manifold has infinitely many totally geodesic hypersurfaces, it must necessarily come from specific arithmetic constructions, by a theorem of Bader--Fisher--Miller--Stover. I will discuss the broader context for these results and some analogies with adjacent questions in dynamical systems and arithmetic geometry.