Abstract: Let (M, g) be a compact, analytic Riemannian manifold with negative curvature. Together with Ben Lowe and Simion Filip, we show that if M contains infinitely many closed totally geodesic hypersurfaces, M is in fact hyperbolic. This means that most negatively curved analytic manifolds have only finitely many closed totally geodesic hypersurfaces.
I will discuss the theorem, some motivations, and some of the ideas in the proof.