Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}{P}^n$

Abstract

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler $n$-dimensional real projective space $(\mathbb{R}{P}^n,F)$ when there exist only finitely many distinct non-contractible closed geodesics on $(\mathbb{R}{P}^n,F)$, where the integer $n\ge 2$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on $\mathbb{R}{P}^n$ with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in \cite{DLX2015} and \cite{Tai2016}, it yields that every $\mathbb{R}{P}^n$ with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.

Advances in Math. to appear

References in corpus (3)

• Multiple closed geodesics on bumpy Finsler $n$-spheres
• Morse theory, closed geodesics, and the homology of free loop spaces
• The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}{P}^n$

Cited by in corpus (1)

• The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form