scholarly journals Cut locus of a left invariant Riemannian metric on SO3 in the axisymmetric case

2016 ◽  
Vol 110 ◽  
pp. 436-453 ◽  
Author(s):  
A.V. Podobryaev ◽  
Yu.L. Sachkov
Keyword(s):  
Riemannian Metric ◽  
Cut Locus ◽  
Axisymmetric Case ◽  
Left Invariant Riemannian Metric

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

scholarly journals C-Flows on a Lie Group for Euler Equations

1970 ◽  
Vol 40 ◽  
pp. 67-84
Author(s):  
Yoshihei Hasegawa
Keyword(s):  
Euler Equations ◽  
Lie Group ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Left Invariant Riemannian Metric ◽  
Invariant Vector Fields

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

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 Randers Ricci soliton homogeneous nilmanifolds

2021 ◽  
Vol 127 (1) ◽  
pp. 100-110
Author(s):  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Vector Field ◽  
Lie Group ◽  
Ricci Flow ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Flow Equation ◽  
Nilpotent Lie Group ◽  
Randers Metric ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected

Let $F$ be a left-invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left-invariant Riemannian metric $\hat{\boldsymbol{a}}$ and a vector field $X$ which is $I_{\hat{\boldsymbol{a}}}(M)$-invariant. We show that if the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton.

scholarly journals The Helgason Fourier transform on a class of nonsymmetric harmonic spaces

1997 ◽  
Vol 55 (3) ◽  
pp. 405-424 ◽  
Author(s):  
Francesca Astengo ◽  
Roberto Camporesi ◽  
Bianca Di Blasio
Keyword(s):  
Fourier Transform ◽  
Riemannian Metric ◽  
Smooth Functions ◽  
One Dimensional ◽  
Symmetric Manifold ◽  
Rank One ◽  
Heisenberg Type ◽  
Solvable Extension ◽  
Left Invariant Riemannian Metric ◽  
The Fourier Transform

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

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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