On a Lie group with constant negative sectional curvature
نویسنده
چکیده
Let λ > 0 be a positive real number, and let n ≥ 1 be an integer. Let G = R×sR be a semi-direct product Lie group where the group multiplication in G is defined by (v1, x1) ∗ (v2, x2) = (v1 + ev2, x1 + x2) for all vi ∈ R, xi ∈ R, and i = 1, 2. We show G has constant sectional curvature −λ, and describe the irreducible unitary representations of G. 2010 Mathematic Subject Classification: 22D10, 22E15
منابع مشابه
Left-invariant Minimal Unit Vector Fields on a Lie Group of Constant Negative Sectional Curvature
We find all left-invariant minimal unit vector fields and strongly normal unit vector fields on a Lie group which is isometric to the hyperbolic space.
متن کاملA remark on left invariant metrics on compact Lie groups
The investigation of manifolds with non-negative sectional curvature is one of the classical fields of study in global Riemannian geometry. While there are few known obstruction for a closed manifold to admit metrics of non-negative sectional curvature, there are relatively few known examples and general construction methods of such manifolds (see [Z] for a detailed survey). In this context, it...
متن کاملACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE
A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...
متن کاملPara-Kahler tangent bundles of constant para-holomorphic sectional curvature
We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...
متن کاملOn Stretch curvature of Finsler manifolds
In this paper, Finsler metrics with relatively non-negative (resp. non-positive), isotropic and constant stretch curvature are studied. In particular, it is showed that every compact Finsler manifold with relatively non-positive (resp. non-negative) stretch curvature is a Landsberg metric. Also, it is proved that every (α,β)-metric of non-zero constant flag curvature and non-zero relatively i...
متن کامل