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Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds

Nicholas Hanson

Abstract

We study non-positively curved closed manifolds $M$ and $n$-dimensional totally geodesic submanifolds of $M \times M$ which satisfy a transversality condition. We prove that, under some mild irreducibility requirements on $M$, if $M \times M$ admits infinitely many such submanifolds or just a single dense such submanifold, then $M$ is a locally symmetric space. In proving this, we prove a stronger version which only requires such submanifolds to exist in the universal cover $\widetilde M \times \widetilde M$.

Totally Geodesic Submanifolds in Products of Non-Positively Curved Manifolds

Abstract

We study non-positively curved closed manifolds and -dimensional totally geodesic submanifolds of which satisfy a transversality condition. We prove that, under some mild irreducibility requirements on , if admits infinitely many such submanifolds or just a single dense such submanifold, then is a locally symmetric space. In proving this, we prove a stronger version which only requires such submanifolds to exist in the universal cover .

Paper Structure

This paper contains 9 sections, 18 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Background
  3. An Idea of the Proof
  4. Definitions and Preliminaries
  5. Proof of the Main Theorem
  6. The General Case
  7. Eliminating Irreducibility Requirements
  8. Generalizing to Arbitrary Products
  9. Other Questions

Key Result

Theorem 1.1

If $\Gamma < \text{SO}_0(n,1)$ is a lattice and the associated space $\text{SO}_0(n,1) / \Gamma$ contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is an arithmetic subgroup. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (30)

  • Theorem 1.1: bader_arithmeticity_2020
  • Theorem 1.2: filip_finiteness_2024
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8: Farb-Weinberger, Theorem 1.2 farb_isometries_2008
  • Theorem 1.9: Farb-Weinberger, Theorem 1.3 farb_isometries_2008
  • Definition 2.1
  • Definition 2.2
  • ...and 20 more