Multiple closed geodesics on 3-spheres

By Y. Long, Huagui Duan. 2009

60 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3c92b8ab7231810e30ebdc791e0c66890e045aac

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In Applied Mathematics

  1. Huagui Duan
  2. Wei Wang
  3. Y. Long
  4. Chun-gen Liu

In Astronomy and Astrophysics

  1. Huagui Duan
  2. Wei Wang
  3. Y. Long

In Geometry and Topology

  1. Huagui Duan
  2. Wei Wang
  3. Y. Long
  4. Chun-gen Liu

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