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Some Rigidity Theorem for Anosov Geodesic Flows

Ítalo Dowell, Sergio Romaña

Abstract

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover, if $M$ has finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity bi-Lipschitz conjugation, and consequently, for $C^1$-conjugacy between two geodesic flows.

Some Rigidity Theorem for Anosov Geodesic Flows

Abstract

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by is of Anosov type, then the constant of contraction of the flow is . Moreover, if has finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity bi-Lipschitz conjugation, and consequently, for -conjugacy between two geodesic flows.

Paper Structure

This paper contains 8 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction and Main Results
  2. Notation and Preliminaries
  3. Geodesic flow
  4. No conjugate points and Ricatti equation
  5. Proof of Theorem \ref{['curvature']}
  6. Rigidity and Lyapunov exponents
  7. Rigidity of Conjugacy
  8. Comparing distances between a manifold and its submanifolds

Key Result

Theorem 1.1

Let $M$ be a complete Riemannian manifold with sectional curvature bounded below by $-c^2$ and Anosov geodesic flow $\phi^{t}_{_M}$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • ...and 21 more