On H-curvature of (α,β)-metrics

Abstract

The non-Riemannian quantity H was introduced by Akbar-Zadeh to characterization of Finsler metrics of constant flag curvature. In this paper, we study two important subclasses of Finsler metrics in the class of so-called (α,β)-metrics, which are defined by F=αϕ(s), s=β/α, where α is a Riemannian metric and β is a closed 1-form on a manifold. We prove that every polynomial metric of degree k≥3 and exponential metric has almost vanishing H-curvature if and only if H=0. In this case, F reduces to a Berwald metric. Then we prove that every Einstein polynomial metric of degree k≥3 and exponential metric satisfies H=0. In this case, F is a Berwald metric.