On a New Class of Weakly Berwald Spaces with ( α, β )-metric
Gauree Shanker *
Centre for Mathematics and Statistics, Central University of Punjab, Bathinda-151001, Punjab, India.
Deepti Choudhary
Department of Mathematics and Statistics, Banasthali University, Banasthali-304022, Rajasthan, India.
*Author to whom correspondence should be addressed.
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.
Keywords: Weakly Berwald Space, second approximate Matsumoto metric, Finsler space, Berwald space, (α, β)-metric.