Riemannian geometry on contact Lie groups

Diatta, André

doi:10.1007/s10711-008-9236-2

Cited by 8 publications

(2 citation statements)

References 19 publications

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“…Equipped with a maximum smooth atlas then we have the notion of a differentiable structure on the topological manifold. The notion of manifolds come in many researches (see for example in reduction theory [8], information of geometrical evaluation [9], application of liquid metal related to manifold [10], symplectic structure on affine Lie group [11] , Poisson Lie group [12], and contact Lie groups [13]). Therefore, we believe that the significance of research in manifolds is very important both in pure and applied mathematics including in non-mathematics research areas.…”

Section: Introductionmentioning

confidence: 99%

A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

Kurniadi

2023

BAREKENG: J. Math. & App.

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There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

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Section: Introductionmentioning

confidence: 99%

A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

Kurniadi

2023

BAREKENG: J. Math. & App.

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“…We will see that some of those examples give rise to contact Lie algebras. See [12], [13], for wider discussions on contact Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

Automorphisms of cotangent bundles of Lie groups

Diatta,

Manga

2008

Preprint

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Let G be a Lie group, G its Lie algebra and T * G its cotangent bundle. On T * G, we consider the Lie group structure obtained by performing a left trivialization and endowing the resulting trivial bundle G × G * with the semi-direct product, using the co-adjoint action of G on the dual space G * of G. We investigate the group of automorphisms of the Lie algebra D := T * G of T * G. More precisely, we fully characterize the Lie algebra of all derivations of D, exhibiting a finer decomposition into components made of well known spaces. Further, we specialize to the cases where G has a bi-invariant Riemannian or pseudo-Riemannian metric, with the semi-simple and compact cases investigated as particular cases.

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A class of Sasakian 5-manifolds

Andrada

Fino

Vezzoni

2009

Transformation Groups

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We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H 2n+1 . Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H 5 or a semidirect product R (H 3 × R). In particular, the compact quotient is an S 1 -bundle over a four-dimensional Kähler solvmanifold.

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Riemannian geometry on contact Lie groups

Cited by 8 publications

References 19 publications

A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

Automorphisms of cotangent bundles of Lie groups

A class of Sasakian 5-manifolds

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