On H-curvature of (α, β)-metrics

Abstract

The non-Riemannian quantity H was introduced by Akbar-Zadeh to characterization of Finsler metrics ofconstant 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-formon a manifold. We prove that every polynomial metric of degree k ≥ 3 and exponential metric has almost vanishingH-curvature if and only if H = 0. In this case, F reduces to a Berwald metric. Then we prove that every Einsteinpolynomial metric of degree k ≥ 3 and exponential metric satisfies H = 0. In this case, F is a Berwald metric.