Related papers: On the left invariant $(\alpha,\beta)$-metrics on …

In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas…

In this paper we study lifted left invariant $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(\alpha,\beta)$-metric of Douglas type $F$, induced by a left invariant…

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…

In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these…

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…

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is…

In this short article, using a left-invariant Randers metric $F$, we define a new left-invariant Randers metric $\tilde{F}$. We show that $F$ is of Berwald (Douglas) type if and only if $\tilde{F}$ is of Berwald (Douglas) type. In the case…

In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(\alpha,\beta)$-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During…

In this paper, we give the flag curvature formula of general $(\alpha,\beta)$-metrics of Berwald type. We study conformally related $(\alpha,\beta)$-metrics, especially general $(\alpha,\beta)$-metrics that are conformally related to…

Let $G$ be a Lie group equipped with a left invariant Randers metric of Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian metrics…

In this paper by using left invariant Riemannian metrics on some 3-dimensional Lie groups we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally…

A left invariant Z-Randers metric on the five-dimensional Heisenberg group is a left invariant Randers metric with deformation vector from the center of the Heisenberg algebra. In this note we prove that for every left invariant Z-Randers…

In this paper we study flag curvature of invariant $(\alpha,\beta)$-metrics of the form $\frac{(\alpha+\beta)^2}{\alpha}$ on homogeneous spaces and Lie groups. We give a formula for flag curvature of invariant metrics of the form…

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…

We prove that every Berwald manifold with non-zero flag curvature is Riemannian. This result provides an extension of Numata and Szabo's rigidity theorems. We show that every positively curved constant isotropic Berwald manifold is…

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex…

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the…

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged…

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(\alpha,\beta)$-metrics. Let $F$ be an $(\alpha,\beta)$-metric which is defined by a left invariant vector field and a left invariant…

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…