In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on the naturally reductive homogeneous manifolds (G/H, g), where the Randers metric induced by the invariant Riemannian metric g and an invariant vector fieldX which is parallel with g.

In this paper we study flag curvature of invariant (α, β)-metrics of the form (α+β) 2 α on homogeneous spaces and Lie groups. We give a formula for flag curvature of invariant metrics of the form F = (α+β) 2 α such that α is induced by an invariant Riemannian metric g on the homogeneous space and the Chern connection of F coincides to the Levi-Civita connection of g. Then some conclusions in the cases of naturally reductive homogeneous spaces and Lie groups are given.

In the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of these metrics. By this study we construct two 4-dimensional Berwald spaces, one of them has non-negative flag curvature and the other one has non-positive flag curvature.

Abstract. In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left invariant Randers metric of Berwald type has three dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.