Publication date:
1 October 2017
Source:Advances in Mathematics, Volume 318
Author(s): Hui Liu, Yuming Xiao
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 [14] and [39], it yields that every $\mathbb{R}{P}^n$ with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.