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Finiteness of totally geodesic hypersurfaces

Simion Filip, David Fisher, Ben Lowe

TL;DR

The paper proves that a closed, real-analytic Riemannian manifold with negative curvature that contains infinitely many closed totally geodesic immersed hypersurfaces must be a hyperbolic manifold that is arithmetic. The approach encodes totally geodesic phenomena as a closed analytic subset in the Grassmann bundle of hyperplanes and then leverages dimension growth, the geodesic and frame flows, and holonomy via Brin-type torsors to force the subset to fill the ambient space. This yields a Cartan-type rigidity statement: the presence of many totally geodesic hypersurfaces enforces constant curvature and arithmetic lattice structure. The work connects classical differential geometry, real-analytic geometry, and homogeneous dynamics, and it outlines both the robustness of the result under finite-volume relaxation and directions for weakening analyticity to smooth settings.

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.

Finiteness of totally geodesic hypersurfaces

TL;DR

The paper proves that a closed, real-analytic Riemannian manifold with negative curvature that contains infinitely many closed totally geodesic immersed hypersurfaces must be a hyperbolic manifold that is arithmetic. The approach encodes totally geodesic phenomena as a closed analytic subset in the Grassmann bundle of hyperplanes and then leverages dimension growth, the geodesic and frame flows, and holonomy via Brin-type torsors to force the subset to fill the ambient space. This yields a Cartan-type rigidity statement: the presence of many totally geodesic hypersurfaces enforces constant curvature and arithmetic lattice structure. The work connects classical differential geometry, real-analytic geometry, and homogeneous dynamics, and it outlines both the robustness of the result under finite-volume relaxation and directions for weakening analyticity to smooth settings.

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.
Paper Structure (40 sections, 17 theorems, 32 equations)

This paper contains 40 sections, 17 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries from dynamics
  3. Outline of section
  4. Anosov flows
  5. Setup
  6. Assumptions for stable/unstable manifolds
  7. Stable/Unstable manifolds
  8. Hopf argument
  9. Center-stable projection
  10. Recurrence and ergodicity in the noncompact case
  11. Torsors and Transitivity Groups
  12. Setup
  13. Reduction of structure group
  14. Stable manifolds on the principal bundle
  15. Holonomy on Principal Bundles
  16. ...and 25 more sections

Key Result

Theorem 1

Let $(M,g)$ be a closed, real-analytic Riemannian manifold with negative sectional curvature, of dimension $n\geq 3$. Suppose that $M$ contains infinitely many closed totally geodesic immersed hypersurfaces. Then $M$ is isometric to a hyperbolic manifold. Furthermore $M$ is arithmetic.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2: Cartan
  • Theorem 3
  • Theorem 4
  • Corollary 1: Open set contains generic point
  • proof
  • Definition 1: right $K$-torsor
  • Remark 1: Properties of the groups and torsors
  • Definition 2: Principal bundle
  • Example 1: Framings
  • ...and 29 more