2010/03/18 by Huagui Duan, Duan, Huagui, Yiming Long +1
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.DS #math.SG
paper · pdf · doi:10.48550/arxiv.1003.3593
59 pages, 1 figure, to appear in Journal of Functional Analysis (JFA) and this is the final version
arxiv created 2010/05/12 · arxiv updated 2010/05/13
In the recent paper \citeLoD1, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic on every irreversible or reversible (including Riemannian) Finsler sphere, and that there exist at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 3-dimensional manifold. In this paper, we study the index growth properties of irrational closed geodesics on Finsler manifolds. This study allows us to extend results in \citeLoD1 on rational and in \citeDuL1, \citeRad4 and \citeRad5 on completely non-degenerate closed geodesics on spheres and \CP2 to every compact simply connected Finsler manifold. Then we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible (including Riemannian) Finsler 4-dimensional manifold.