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

Abstract

We show that a compact C ∞ C^{\infty } , k-basic Finsler surface without conjugate points and genus greater than one is Riemannian. This result is a C ∞ 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 ∞ 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.