Classification of Douglas (α,β)-metrics on five-dimensional nilpotent Lie groups

In this paper, we classify all simply connected five-dimensional nilpotent Lie groups which admit [Formula: see text]-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification, we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and [Formula: see text]-curvature.

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ON THE RANDERS METRICS ON TWO-STEP HOMOGENEOUS NILMANIFOLDS OF DIMENSION FIVE

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887811005257 ◽

2011 ◽

Vol 08 (03) ◽

pp. 501-510 ◽

Cited By ~ 4

Author(s):

HAMID REZA SALIMI MOGHADDAM

Keyword(s):

Scalar Curvature ◽

Sectional Curvature ◽

Lie Groups ◽

Three Dimensional ◽

Flag Curvature ◽

Nilpotent Lie Groups ◽

Randers Metric ◽

Levi Civita Connection ◽

Simply Connected ◽

Randers Metrics

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

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C-Flows on a Lie Group for Euler Equations

Nagoya Mathematical Journal ◽

10.1017/s0027763000013866 ◽

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.

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On the geometrical properties of hypercomplex four-dimensional Lie groups

Georgian Mathematical Journal ◽

10.1515/gmj-2018-0003 ◽

2020 ◽

Vol 27 (1) ◽

pp. 111-120 ◽

Cited By ~ 1

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.

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

Yugra State University Bulletin ◽

10.17816/byusu20210123-29 ◽

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.

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Three-dimensional solvsolitons and the minimality of the corresponding submanifolds

International Journal of Mathematics ◽

10.1142/s0129167x17500483 ◽

2017 ◽

Vol 28 (06) ◽

pp. 1750048 ◽

Cited By ~ 2

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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Hardy's Theorem for simply connected nilpotent Lie groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005424 ◽

2001 ◽

Vol 131 (3) ◽

pp. 487-494 ◽

Cited By ~ 15

Author(s):

EBERHARD KANIUTH ◽

AJAY KUMAR

Keyword(s):

Fourier Transform ◽

Lie Groups ◽

Nilpotent Lie Groups ◽

Heisenberg Groups ◽

Hardy’S Theorem ◽

Simply Connected

We prove an analogue of Hardy's Theorem for Fourier transform pairs in ℝ for arbitrary simply connected nilpotent Lie groups, thus extending earlier work on ℝn and the Heisenberg groups ℍn.

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