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.

On the Randers metrics on two-step homogeneous nilmanifolds of dimension five

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.

On the randers metrics on two-step homogeneous nilmanifolds of dimension five

Some Berwald spaces of non-positive flag curvature

In the present paper we study Randers metrics of Berwald type on simply connected four-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 formulae for computing the flag curvature of these metrics. By this study we construct two four-dimensional Berwald spaces, one of them with a non-negative flag curvature and the other one with a non-positive flag curvature.

Randers metrics of Berwald type on four-dimensional hypercomplex Lie groups

In this paper, we study the flag curvature of invariant (α, β)-metrics of the form on homogeneous spaces and Lie groups. We give a formula for the flag curvature of invariant metrics of the form 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.

The flag curvature of invariant (α, β)-metrics of type \frac{(\alpha+\beta)^2}{\alpha}

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.

On the Flag Curvature of Invariant Randers Metrics

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

On some hypercomplex 4-dimensional lie groups of constant scalar curvature

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invariant vector field, in a special case, the Lie group admits a left invariant Randers metric such that this intersection is a maximal compact subgroup of the group of automorphisms with respect to which the considered vector field is invariant.

Hamid Reza Salimi Moghaddam

Papers

On the Flag Curvature of Invariant Randers Metrics

On the randers metrics on two-step homogeneous nilmanifolds of dimension five

On some hypercomplex 4-dimensional lie groups of constant scalar curvature

The Relation Between Automorphism Group and Isometry Group of Randers Lie Groups