Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

TL;DR

This paper extends the Cheeger-Gromoll-Lichnerowicz splitting theorem to Finsler manifolds with nonnegative Ricci curvature, demonstrating measure-preserving and isometric splitting under certain conditions, and provides topological estimates.

Contribution

It generalizes classical splitting results to Finsler geometry, including measure-preserving and isometric splitting for specific Berwald spaces, and offers Betti number estimates.

Findings

Finsler manifolds with nonnegative Ricci curvature admit a measure-preserving splitting.

Special Berwald spaces allow for an isometric splitting generated by gradient vector fields.

A Betti number estimate is established for Berwald spaces.

Abstract

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.