Generalized Berwald spaces with (α,β)-metrics

In this paper, we are going to study generalized Berwald manifold with ( α , β ) -metrics. We show that a Finsler manifold with ( α , β ) -Finsler function of sign property is a generalized Berwald manifold if and only if there exists a covariant derivative such that it is compatible with α and β and equivalently if and only if the dual vector field β ♯ is of constant Riemannian length. This generalizes the result previously only known in the case of Randers manifold. Then, we find the necessary and sufficient condition under which an ( α , β ) -metric of sign property is compatible with a semi-symmetric covariant derivative. In the following, we consider the β -change of reversible Finsler manifolds and find some conditions under which these manifolds are generalized Berwaldian. Finally, we study the effect of our theory on left invariant Finsler function on Lie groups.