Resonance identity and multiplicity of non-contractible closed geodesics on Finsler ℝPⁿ

2016/07/10 by Liu, Hui, Xiao, Yuming
#53C22 #58E05 #58E10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1607.02746

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 (ℝPⁿ,F) when there exist only finitely many distinct non-contractible closed geodesics on (ℝPⁿ,F), where the integer n≥2. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on ℝPⁿ with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in \citeDLX2015 and \citeTai2016, it yields that every ℝPⁿ with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.

Related

• The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
• Multiplicity of closed geodesics on bumpy Finsler manifolds with elliptic closed geodesics
• The Fadell-Rabinowitz index and multiplicity of non-contractible closed geodesics on Finsler ℝPⁿ
• Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres
• Multiplicity of non-contractible closed geodesics on Finsler compact space forms