On a New Class of Weakly Berwald Spaces with ( α, β )-metric

Abstract

We have two concepts of Douglas spaces and Landsberg spaces as generalizations of Berwald spaces. S. Bacso [1] gave the definition of a weakly-Berwald space as another generalization of Berwald spaces. In 1972, M. Matsumoto has introduced the concept of (α, β)-metric, which is a Finsler meric, contstructed from a Riemannian metric and a differential 1-form. In this paper, we study an important class of (α, β)-metrics in the form L =   , known as second approximate Matsumoto metric on an n-dimensional manifold and get the conditions for such metrics to be weakly-Berwald metrics, where α =   is a Riemannian metric and β = biyi is a 1-form. A Finsler space with an (α,β)-metric is a weakly-Berwald space, if and only if  is a 1-form. We show that it becomes a weakly Berwald space under some geometric and algebraic conditions.