Weinstein's theorem for Finsler manifolds
An isometry of a Finsler manifold is a diffeomorphism \(f: M\to M \) which preserves the distance. Equivalently, \(f\) preserves the Finsler norms of the tangent vectors cf. [\textit{S. Deng} and \textit{Z. Hou}, Pac. J. Math. 207, No.~1, 149--155 (2002; Zbl 1055.53055)]. The authors prove Weinstein's Theorem for Finsler manifolds: Let \(f\) be an isometry of a compact oriented positively homogeneous Finsler manifold \(M\) of dimension \(n\). If \(M\) has positive flag curvature and \(f\) preserves the orientation of \(M\) for \(n\) even and reverses the orientation of \(M\) for \(n\) odd, then, \(f\) has a fixed point. The proof is very detailed, based on results from the Finslerian geometry, on the line of Weinstein's proof.
scientific article; zbMATH DE number 5128994