A first integral for $C^{\infty }$, k-basic Finsler surfaces and applications to rigidity
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- by P. Foulon and R. Ruggiero PDF
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Abstract:
We show that a compact $C^{\infty }$, k-basic Finsler surface without conjugate points and genus greater than one is Riemannian. This result is a $C^{\infty }$ version of the fact, proved by G. Paternain, that analytic, compact, k-basic Finsler surfaces with genus greater than one are Riemannian. The proof in the $C^{\infty }$ case relies mainly on two facts: first of all the existence of a first integral for the geodesic flow of any k-basic Finsler surface, one of the main contributions of this note; and secondly the triviality of every first integral assuming the absence of conjugate points.References
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Additional Information
- P. Foulon
- Affiliation: Centre International de Rencontres Mathématiques-CIRM, 163 avenue de Luminy, Case 916, F-13288 Marseille - Cedex 9, France
- MR Author ID: 68355
- Email: foulon@cirm-mathl.fr
- R. Ruggiero
- Affiliation: Departamento de Matemática, PUC-Rio, Rua Marqués de São Vicente 225, Rio de Janeiro, Brazil, 22453-900
- MR Author ID: 313673
- Email: rorr@mat.puc-rio.br
- Received by editor(s): January 28, 2015
- Received by editor(s) in revised form: October 28, 2015
- Published electronically: March 17, 2016
- Additional Notes: The second author was partially supported by CNPq, CAPES, FAPERJ and CIRM
- Communicated by: Yingfei Yi
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3847-3858
- MSC (2010): Primary 37D40, 58B20; Secondary 53D25, 53C24
- DOI: https://doi.org/10.1090/proc/13079
- MathSciNet review: 3513543