Properties of Berwald scalar curvature

TL;DR

This paper proves that Finsler manifolds with zero Berwald scalar curvature have zero E-curvature, and explores implications for Landsberg and Berwald manifolds, especially for (,eta)-metrics, improving previous results.

Contribution

It establishes new links between Berwald scalar curvature, E-curvature, and Landsberg properties, extending understanding of Finsler geometry.

Findings

Vanishing Berwald scalar curvature implies zero E-curvature.

Landsberg manifolds with zero Berwald scalar curvature are Berwald manifolds.

For (,eta)-metrics, vanishing mean Landsberg and Berwald scalar curvatures imply vanishing Berwald curvature.

Abstract

In this short paper, we prove that a Finsler manifold with vanishing Berwald scalar curvature has zero $\mathbf{E}$-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. This improves a previous result in \cite{Li}. For $(\alpha,\beta)$-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, the Berwald curvature also vanishes.