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Non-contractible closed geodesics on compact Finsler space forms without self-intersections

Yuchen Wang

Abstract

Let $M=S^n/ Γ$ and $h \in π_1(M)$ be a non-trivial element of finite order $p$, where the integers $n, p\geq2$ and $Γ$ is a finite abelian group which acts on the sphere freely and isometrically, therefore $M$ is diffeomorphic to a compact space form which is typical a non-simply connected manifold. We prove there exist at least two non-contractible closed geodesics on $\mathbb{R}P^2$ and obtain the upper bounds on their lengths. Moreover, we prove there exist at least $n$ prime non-contractible simple closed geodesics on $(M,F)$ of prescribed class $[h]$, provided \[ F^2 <(\frac{λ+1}λ)^2 g_0 \;\; \text{ and } \;\; (\fracλ{λ+1})^2 < K \leq 1 \text{ for $n$ is odd or }\; 0<K \leq 1 \text{ for $n$ is even}, \] where $λ$ is the reversibility, $K$ is the flag curvature and $g_0$ is standard Riemannian metric. Stability of these non-contractible closed geodesics is also studied.

Non-contractible closed geodesics on compact Finsler space forms without self-intersections

Abstract

Let and be a non-trivial element of finite order , where the integers and is a finite abelian group which acts on the sphere freely and isometrically, therefore is diffeomorphic to a compact space form which is typical a non-simply connected manifold. We prove there exist at least two non-contractible closed geodesics on and obtain the upper bounds on their lengths. Moreover, we prove there exist at least prime non-contractible simple closed geodesics on of prescribed class , provided where is the reversibility, is the flag curvature and is standard Riemannian metric. Stability of these non-contractible closed geodesics is also studied.
Paper Structure (7 sections, 13 theorems, 65 equations)

This paper contains 7 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morse theory of non-contractible closed geodesics
  4. Index theory of symplectic patch and precise iteration formula
  5. Geometric properties concern Finsler manifolds
  6. Existence of non-contractible closed geodesics on $\mathbb{R}P^n$
  7. Equivariant Morse theory for non-contractible closed geodesics

Key Result

Theorem 1.1

There exist at least two distinct non-contractible closed geodesics on $(\mathbb{R}P^2, F)$ with reversibility $\lambda$ and the flag curvature $K$ satisfying $\left(\frac{\lambda}{\lambda+1} \right)^2<\delta \leq K \leq 1$, where $c_1$ is the minimal energy closed curve on $\Lambda_h M$ with

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Theorem 3 of Tai16 or Lemma 2.3 of LLX18
  • Proposition 2.2
  • Lemma 2.3: Theorem I.4.3 of Chan93
  • Lemma 2.4: Theorem 3.1 of LX
  • Lemma 2.5: cf. Lemma 1 in Rad07
  • Lemma 2.6: cf. Theorem 4 and 5 in Rad04
  • Lemma 2.7
  • ...and 10 more