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Riemannian manifolds with Anosov geodesic flow do not have conjugate points

Ítalo Melo, Sergio Romaña

Abstract

This paper establishes a significant result concerning the absence of conjugate points in certain complete Riemannian manifolds. Specifically, we demonstrate that any complete non-compact manifold with curvature bounded below and an Anosov geodesic flow does not possess conjugate points. This resolves an open problem left by R. Mañé in [9] and subsequently highlighted by [7].

Riemannian manifolds with Anosov geodesic flow do not have conjugate points

Abstract

This paper establishes a significant result concerning the absence of conjugate points in certain complete Riemannian manifolds. Specifically, we demonstrate that any complete non-compact manifold with curvature bounded below and an Anosov geodesic flow does not possess conjugate points. This resolves an open problem left by R. Mañé in [9] and subsequently highlighted by [7].

Paper Structure

This paper contains 13 sections, 28 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Paper Structure:
  3. Notation and Basic Concepts
  4. Geodesic flow
  5. Jacobi fields and the differential of the geodesic flow
  6. Symplectic geometry
  7. Graphs and Riccati equation
  8. Classical results
  9. The intersection between the vertical subspace and $E^{s,u}$
  10. Properties of sets $\mathcal{B}^{s}$ and $\mathcal{B}^u$
  11. Topological properties of the first positive (negative) time to $\mathcal{B}^{s,u}$
  12. Closedness of $\mathcal{B}^{u}_{+}$ and $\mathcal{B}^s_{-}$
  13. Proof of the Main Results

Key Result

Theorem 1.1

Suppose $M$ is a non-compact two-dimensional manifold with sectional curvature bounded below. If the geodesic flow of $M$ is Anosov, then $M$ has no conjugate points.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 54 more