Finiteness of totally geodesic hypersurfaces

TL;DR

This paper proves that closed negatively curved analytic Riemannian manifolds contain only finitely many totally geodesic hypersurfaces unless they are of constant curvature, linking geometric structure to arithmetic properties.

Contribution

It establishes a finiteness result for totally geodesic hypersurfaces in negatively curved manifolds, showing they are finite unless the manifold has constant curvature.

Findings

Manifolds with infinitely many totally geodesic hypersurfaces are isometric to arithmetic hyperbolic manifolds.

Finitely many such hypersurfaces imply the manifold does not have constant curvature.

The result connects geometric properties with arithmetic and curvature conditions.

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.