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Multiple closed geodesics on 3-spheres
- Source :
-
Advances in Mathematics . Aug2009, Vol. 221 Issue 6, p1757-1803. 47p. - Publication Year :
- 2009
-
Abstract
- 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 Poincaré map contains no 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 , 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. [Copyright &y& Elsevier]
- Subjects :
- *GEODESICS
*DIFFERENTIAL geometry
*GLOBAL analysis (Mathematics)
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 00018708
- Volume :
- 221
- Issue :
- 6
- Database :
- Academic Search Index
- Journal :
- Advances in Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 40120904
- Full Text :
- https://doi.org/10.1016/j.aim.2009.03.007