In this paper we study lifted left invariant $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a group equipped with $(\alpha,\beta)$-metric $F$, induced by Riemannian metric $g$. Using vertical and complete lifts, construct the $F^v$ $F^c$ $TG$ give necessary sufficient conditions for them to type. Then, flag curvature these metrics are studied. Finally, as some specia...

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. algebras with a codimension-one ideal. In particular, we classify six-dimensional which carry structure. It has been conjectured in [1] that compact complex manifold admitting both metric and an SKT necessarily Kähler metric: prove this conjecture for solvmanifolds left-invariant structures. Moreover, investigate the be...

in this paper, we investigate the concept of topological stationary for locally compact semigroups. in [4], t. mitchell proved that a semigroup s is right stationary if and only if m(s) has a left invariant mean. in this case, the set of values ?(f) where ? runs over all left invariant means on m(s) coincides with the set of constants in the weak* closed convex hull of right translates of f. th...

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie dimension three. As an application, we determine index symmetry such metrics and prove that singularities moduli space metrics, up to isometric automorphism are contained in subspace classes with maximal symmetry.

Timelike surfaces in the three-dimensional Heisenberg group with left-invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal characterized by non-conformal Lorentz harmonic maps into de Sitter two sphere. On basis of characterization, generalized Weierstrass type representation will be established through loop decompositions.

Let G be a Poisson–Lie group equipped with left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on tangent bundle TG of G. In this paper, we induce metric TG, and express in different cases Levi-Civita connection curvature terms We prove that space differential forms Ω*(G) is graded algebra if, only Ω*(TG) algebra. Moreover, show Sanchez de Al...

We consider simply connected, 2-step nilpotent Lie groups N, all of which are diffeomorphic to Euclidean spaces via the Lie group exponential map exp : ˆ → N. We show that every such N with a suitable left invariant metric is the base space of a Riemannian submersion ρ : N* → N, where the fibers of ρ are flat, totally geodesic Euclidean spaces. The left invariant metric and Lie algebra of N* ar...

we present some new results on the existence and the approximationof common fixed point of expansive mappings and semigroups in probabilisticmetric spaces.