Properties of Berwald scalar 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.

arXiv commentpublished on Front. Math. China