curvature

This paper is devoted to study of a class of conformally flat (α,β)-metrics that have of the form F = αexp(2s)/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.

In this paper, we study generalized Douglas-Weyl Finsler metrics. We find some conditions under which the class of generalized Douglas-Weyl (α, β)-metric with vanishing S-curvature reduce to the class of Berwald metrics.

We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman''s and Skrypnik''s variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik''s method, described in cite{vyr3}. In local coordinates, equilibrium points satisfy an elliptic boundary value problem.