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A quantitative closing Lemma and Partner orbits on Riemannian manifolds with negative curvature

Michela Egidi, Gerhard Knieper

Abstract

In this paper we prove a quantitative closing Lemma for manifolds of negative sectional curvature. As an application we study partner and pseudo-partner orbits for self-crossing closed geodesic.

A quantitative closing Lemma and Partner orbits on Riemannian manifolds with negative curvature

Abstract

In this paper we prove a quantitative closing Lemma for manifolds of negative sectional curvature. As an application we study partner and pseudo-partner orbits for self-crossing closed geodesic.
Paper Structure (9 sections, 26 theorems, 190 equations, 5 figures)

This paper contains 9 sections, 26 theorems, 190 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic facts on Hadamard manifolds
  3. Boundary at infinity
  4. Busemann functions
  5. Some trigonometrical estimates for pinched negatively curved Hadamard manifolds
  6. A quantitative closing lemma
  7. Construction of partner orbits
  8. Construction of partner orbits for surfaces
  9. Construction of pseudo-partner orbits

Key Result

Theorem 1

There exist $\delta_0=\delta_0(\kappa_1,\kappa_2)\in(0,\frac{1}{2})$ and $t_0=t_0(\kappa_1,\kappa_2)>1$ such that for all $\delta\leq \delta_0$, all $T\geq t_0$ and all $w\in SM$ with $d_1(w,\phi^T(w))\leq \delta$ there exist $u\in SM$, $T'>0$, with $\phi^{T'}(u)=u$ and where the constant $C$ is given by Furthermore,

Figures (5)

  • Figure 1.1: Geodesic $c_w$ with self-crossing with small crossing angle $\varepsilon =\sphericalangle_p (w, -\dot c_w(T_1))$ and its partner orbit $c_u$.
  • Figure 1.2: Geodesic $c_w$ with a self-crossing of large crossing angle $\sphericalangle_p (w, -\dot c_w(T_1)) \in [\pi -\varepsilon, \pi]$ and its pseudo-partner orbit.
  • Figure 2.1: The $4$-gons in Lemma \ref{['lem:perp-proj']}.
  • Figure 3.1: Proving the inclusion $\gamma \mathbf{F}_{\rho(\theta,t)} \subset \mathbf{F}_{\rho(\theta,t)}$ in proposition \ref{['prop:c']}.
  • Figure 4.1: Proving Theorem \ref{['thm:partner-general']}.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 42 more