Some Rigidity Results on Complete Finsler Manifolds

TL;DR

This paper extends classical rigidity theorems to Finsler geometry, showing that complete Finsler manifolds with positive constant flag curvature are essentially spherical, and classifies manifolds admitting certain transnormal functions.

Contribution

It generalizes Obata's theorem to Finsler manifolds and provides a classification of manifolds with specific transnormal functions.

Findings

Complete Finsler manifolds with positive constant flag curvature are homeomorphic to spheres.

Such manifolds are isometrically homeomorphic to Euclidean spheres with special Finsler metrics.

Manifolds admitting transnormal functions with two critical points are topologically spheres.

Abstract

We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second order differential equation. Mainly, we prove that every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an Euclidean sphere endowed with a certain Finsler metric and vice versa. Based on these results, we present a classification of Finsler manifolds which admit a transnormal function. Specifically, we show that if a complete Finsler manifold admits a transnormal function with exactly two critical points, then it is homeomorphic to a sphere.