A first integral for \(C^\infty\), \(k\)-basic Finsler surfaces and applications to rigidity (Q2814405)

From the text: 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 \textit{G. P. Paternain} [Houston J. Math. 23, No. 3, 421--426 (1997; Zbl 0896.53046)], that analytic, compact, \(k\)-basic Finsler surfaces with genus greater than one are Riemannian. The proof in the \( C^{\infty }\) case mainly relies 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.