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

We have two concepts of Douglas spaces and Landsberg spaces as generalizations of Berwaldspaces. S. Bacso [1] gave the definition of a weakly-Berwald space as another generalization ofBerwald spaces. In 1972, M. Matsumoto has introduced the concept of (α, β)-metric, which is aFinsler 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 formL=α+β+2+32, known as secondapproximate Matsumoto metric on an n-dimensional manifold and get the conditions for suchmetrics to be weakly-Berwald metrics, whereα=√aijyiyjis a Riemannian metric andβ=biyiis a 1-form. A Finsler space with an (α, β)-metric is a weakly-Berwald space, if and only ifBmm=∂Bm/∂ymis a 1-form. We show that it becomes a weakly Berwald space under somegeometric and algebraic conditions.