Some remarks on Myers theorem for Finsler manifolds

TL;DR

This paper extends the Bonnet-Myers theorem to Finsler manifolds by establishing conditions on the average Ricci scalar that imply compactness and diameter bounds, using index form methods.

Contribution

It introduces new conditions on the average Ricci scalar for Finsler manifolds that ensure compactness and diameter bounds, generalizing classical results.

Findings

Finsler manifolds are compact under certain average Ricci scalar bounds.

An upper diameter bound is established when Ricci scalar is bounded above.

Compactness can be achieved without explicit Ricci scalar bounds, using average conditions.

Abstract

The standard Bonnet-Myers theorem says that if the Ricci scalar of a Riemannian manifold is bounded below by a positive number, then the manifold is compact. Moreover, a bound of its diameter is pointed out. The theorem was extended to Finsler manifolds. In this paper we prove that if a certain condition on the average of the Ricci scalar holds, then the Finsler manifold M is compact if the Ricci scalar is bounded above by the same positive number. An upper bound of the diameter is also found. With no condition on Ricci scalar itself but with a different one on its average, we find that the Finsler manifold M is again compact. This time no bound of the diameter is found.The proofs are given in the Finslerian setting and are based on the index form along geodesics.