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Symmetries of geodesic flows on covers and rigidity

Daniel Mitsutani

Abstract

We define and study the foliated centralizer: the group of $C^\infty$ centralizer elements of the lift of an Anosov system on a non-compact manifold which additionally preserve the stable and unstable foliations. When the Anosov system is the geodesic flow of a closed Riemannian manifold with pinched negative sectional curvatures, we prove some rigidity properties for the foliated centralizer of the lift of the flow to the universal cover: it is a finite-dimensional Lie group, which is moreover discrete (modulo the action of the flow itself) unless the metric is homothetic to some real hyperbolic metric. This result is inspired by the study of isometries of universal covers that appeared originally in the work of Eberlein, and later in Farb and Weinberger, as well as centralizer rigidity results in dynamics.

Symmetries of geodesic flows on covers and rigidity

Abstract

We define and study the foliated centralizer: the group of centralizer elements of the lift of an Anosov system on a non-compact manifold which additionally preserve the stable and unstable foliations. When the Anosov system is the geodesic flow of a closed Riemannian manifold with pinched negative sectional curvatures, we prove some rigidity properties for the foliated centralizer of the lift of the flow to the universal cover: it is a finite-dimensional Lie group, which is moreover discrete (modulo the action of the flow itself) unless the metric is homothetic to some real hyperbolic metric. This result is inspired by the study of isometries of universal covers that appeared originally in the work of Eberlein, and later in Farb and Weinberger, as well as centralizer rigidity results in dynamics.
Paper Structure (19 sections, 12 theorems, 25 equations)

This paper contains 19 sections, 12 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Hidden Symmetries.
  3. Rigid Geometric Structures.
  4. Outline of the Proof.
  5. Acknowledgements
  6. Preliminaries
  7. Topological Transformation Groups
  8. $C^r$-topologies
  9. Montgomery-Zippin's solution to Hilbert's Fifth Problem
  10. $C^r$-foliations
  11. Foliations with continuously $C^r$-leaves
  12. Connections along foliations
  13. Journé's Theorem
  14. Geodesic Flows in Negative Curvature
  15. Geodesic Flows
  16. ...and 4 more sections

Key Result

Theorem A

For $n \geq 3$, let $(M^n,g)$ be a closed Riemmanian manifold with negative strictly $\frac{1}{4}$-pinched sectional curvatures, that is, with sectional curvatures $-4 < K \leq -1$. If the reduced foliated centralizer $G_c$ is non-trivial, then $(M,g)$ is homothetic to a real hyperbolic manifold.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem A
  • Remark 1.2
  • Theorem B
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5: Fiber Bunching
  • ...and 19 more