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On Finsler surfaces without conjugate points

Published online by Cambridge University Press:  07 February 2012

JOSÉ BARBOSA GOMES  and
RAFAEL O. RUGGIERO
JOSÉ BARBOSA GOMES
Affiliation:
Departamento de Matemática, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, 36036-330, Brazil (email: barbosa.gomes@ufjf.edu.br)
RAFAEL O. RUGGIERO
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22453-900, Brazil (email: rorr@mat.puc-rio.br)
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Abstract

If (M,F) is a C4 compact Finsler surface of genus at least two without conjugate points, we show that the first integrals of the geodesic flow are constant. Using this fact, we show that if (M,F) is also of Landsberg type then (M,F) is Riemannian. The connection between the absence of conjugate points and the Riemannian character of the Finsler metric has some remarkable consequences concerning rigidity.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 455 - 474
Copyright
©2012 Cambridge University Press

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