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A first integral for $ C^{\infty}$, k-basic Finsler surfaces and applications to rigidity

Authors: P. Foulon and R. Ruggiero
Journal: Proc. Amer. Math. Soc. 144 (2016), 3847-3858
MSC (2010): Primary 37D40, 58B20; Secondary 53D25, 53C24
Published electronically: March 17, 2016
MathSciNet review: 3513543
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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.

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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

R. Ruggiero
Affiliation: Departamento de Matemática, PUC-Rio, Rua Marqués de São Vicente 225, Rio de Janeiro, Brazil, 22453-900

Keywords: Finsler surface, k-basic Finsler manifolds, conjugate points
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
Article copyright: © Copyright 2016 American Mathematical Society

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