Uniform controllable sets of left-invariant vector fields on noncompact Lie groups

1986 ◽  
Vol 7 (3) ◽  
pp. 213-216 ◽  
Author(s):  
F. Silva Leite
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

scholarly journals Transitivity of families of invariant vector fields on the semidirect products of Lie groups

1982 ◽  
Vol 271 (2) ◽  
pp. 525-525 ◽  
Author(s):  
B. Bonnard ◽  
V. Jurdjevic ◽  
I. Kupka ◽  
G. Sallet
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Invariant Vector ◽  
Semidirect Products ◽  
Invariant Vector Fields

Left-Invariant Vector Fields

2013 ◽  
pp. 45-49 ◽  
Author(s):  
Daniel Bump
Keyword(s):  
Vector Fields ◽  
Invariant Vector ◽  
Left Invariant Vector ◽  
Invariant Vector Fields

On the flag curvature of invariant (α,β)-metrics

2016 ◽  
Vol 13 (04) ◽  
pp. 1650039 ◽  
Author(s):  
M. Parhizkar ◽  
D. Latifi
Keyword(s):  
Lie Groups ◽  
Explicit Formula ◽  
Vector Fields ◽  
Homogeneous Spaces ◽  
Riemannian Metrics ◽  
Flag Curvature ◽  
Invariant Vector ◽  
Invariant Riemannian Metrics ◽  
Randers Metrics ◽  
Invariant Vector Fields

In this paper, we consider invariant [Formula: see text]-metrics which are induced by invariant Riemannian metrics [Formula: see text] and invariant vector fields [Formula: see text] on homogeneous spaces. We study the flag curvatures of invariant [Formula: see text]-metrics. We first give an explicit formula for the flag curvature of invariant [Formula: see text]-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.

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.

On the geometrical properties of hypercomplex four-dimensional Lie groups

2020 ◽  
Vol 27 (1) ◽  
pp. 111-120 ◽  
Author(s):  
Mehri Nasehi ◽  
Mansour Aghasi
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Invariant Vector ◽  
Geometrical Properties ◽  
Invariant Vector Field ◽  
Left Invariant Vector ◽  
Totally Geodesic Hypersurfaces ◽  
Yamabe Solitons

AbstractIn this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

scholarly journals The Number of Sides of a Parallelogram

1999 ◽  
Vol Vol. 3 no. 2 ◽  
Author(s):  
Elisha Falbel ◽  
Pierre-Vincent Koseleff
Keyword(s):  
Nilpotent Group ◽  
Lie Group ◽  
Vector Fields ◽  
Invariant Vector ◽  
Free Nilpotent Group ◽  
Rational Series ◽  
Left Invariant Vector ◽  
International Audience ◽  
Invariant Vector Fields

International audience We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series.

scholarly journals One-harmonic invariant vector fields on three-dimensional Lie groups

2012 ◽  
Vol 62 (6) ◽  
pp. 1532-1547 ◽  
Author(s):  
E. Calviño-Louzao ◽  
J. Seoane-Bascoy ◽  
M.E. Vázquez-Abal ◽  
R. Vázquez-Lorenzo
Keyword(s):  
Lie Groups ◽  
Vector Fields ◽  
Three Dimensional ◽  
Invariant Vector ◽  
Invariant Vector Fields
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