Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric

2014 ◽  
Vol 55 (5) ◽  
pp. 783-791 ◽  
Author(s):  
V. N. Berestovskĭ
Keyword(s):  
Lie Groups ◽  
Riemannian Metric

Characteristic functions of semigroups in semi-simple Lie groups

2019 ◽  
Vol 31 (4) ◽  
pp. 815-842
Author(s):  
Luiz A. B. San Martin ◽  
Laercio J. Santos
Keyword(s):  
Laplace Transform ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Convex Cone ◽  
Convex Set ◽  
Flag Manifold ◽  
Riemannian Metric ◽  
Iwasawa Decomposition ◽  
Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

scholarly journals LEFT INVARIANT $(\alpha,\beta)$-METRICS ON 4-DIMENSIONAL LIE GROUPS

2020 ◽  
pp. 727
Author(s):  
Mona Atashafrouz ◽  
Behzad Najafi ◽  
Laurian-Ioan Piscoran
Keyword(s):  
Real Number ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Metric ◽  
Flag Curvature ◽  
Hypercomplex Structure ◽  
Alpha Beta ◽  
Beta 2 ◽  
Positive Flag Curvature

Let $G$ be a 4-dimensional Lie group with an invariant para-hypercomplex structure and let $F= \beta+ a\alpha+\beta^2/{\alpha}$ be a left invariant $(\alpha,\beta)$-metric, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on $G$, and $a$ is a real number. We prove that the flag curvature of $F$ with parallel 1-form $\beta$ is non-positive, except in Case 2, in which $F$ admits both negative and positive flag curvature. Then, we determine all geodesic vectors of $(G,F)$.  

On harmonic tensors on three-dimensional Lie groups with left-invariant Riemannian metric

2008 ◽  
Vol 77 (2) ◽  
pp. 306-309 ◽  
Author(s):  
O. P. Gladunova ◽  
E. D. Rodionov ◽  
V. V. Slavskii
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Left Invariant Riemannian Metric

XIX.—Metrisable Lie Groups and Algebras

1957 ◽  
Vol 64 (3) ◽  
pp. 290-304 ◽  
Author(s):  
S.-T. Tsou ◽  
A. G. Walker
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Necessary Conditions ◽  
Existence Theorems ◽  
Riemannian Metric ◽  
Abelian Algebra ◽  
Lie Groups And Algebras ◽  
Complex Extension

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

scholarly journals Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

2021 ◽  
Vol 60 (1) ◽  
pp. 23-29
Author(s):  
Pavel N. Klepikov ◽  
Evgeny D. Rodionov ◽  
Olesya P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Natural Generalization ◽  
Tensor Fields ◽  
Parallel Transfer ◽  
Left Invariant Riemannian Metric

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

scholarly journals Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

2017 ◽  
Vol 28 (06) ◽  
pp. 1750048 ◽  
Author(s):  
Takahiro Hashinaga ◽  
Hiroshi Tamaru
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
Solvable Lie Groups ◽  
Left Invariant Riemannian Metric ◽  
Invariant Riemannian Metrics ◽  
Simply Connected

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

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