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Multiple closed geodesics on 3-spheres
- Source :
- Advances in Mathematics. (6):1757-1803
- Publisher :
- Elsevier Inc.
-
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.
- 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
Subjects
Details
- Language :
- English
- ISSN :
- 00018708
- Issue :
- 6
- Database :
- OpenAIRE
- Journal :
- Advances in Mathematics
- Accession number :
- edsair.doi.dedup.....afea196806a9b742a07d230e48b5b8a9
- Full Text :
- https://doi.org/10.1016/j.aim.2009.03.007