On a class of conformally flat (α,β)-metrics with special curvature properties

This paper is devoted to study of a class of conformally flat (α,β)-metrics that have of the form F = αexp(۲s)/s; where s := β/α. They are called Kropina change of exponential (α,β)-metrics. We prove that if F has relatively isotropic mean Landsberg curvature or almost vanishing Xi-curvature then it is a Riemannian metric or a locally Minkowski metric. Also, we prove that, if F be a weak Einstein metric, then it is either a Riemannian metric or a locally Minkowski metric.

Keywords:

Conformally flat metric , exponential (alpha , beta)-metric , Xi-curvature , mean Landsberg curvature