1. Multiple closed geodesics on 3-spheres
- Author
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Huagui Duan and Yiming Long
- Subjects
Mathematics(all) ,Pure mathematics ,Geodesic ,Rational closed geodesics ,General Mathematics ,Geodesic map ,Mathematical analysis ,Rotation matrix ,Prime (order theory) ,Closed geodesic ,Simply connected space ,Mathematics::Metric Geometry ,Finsler manifold ,Mathematics::Differential Geometry ,Homological method ,3-spheres ,Rotation (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113–149]) of its linearized Poincare map contains no 2 × 2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d ⩾ 2 , it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.
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