Homogeneous Invariant Ricci Solitons on Four-dimensional Lie Groups

Клепиков, Павел Николаевич; Oskorbin, D. N.

doi:10.14258/izvasu(2015)1.2-21

Cited by 6 publications

(5 citation statements)

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“…The condition αγ = 0 yields α = 0. The first and the fourth equations of (13) imply that γ = 0, which is a contradiction. Therefore, Lorentzian non-unimodular Lie groups do not accept any left-invariant cross curvature soliton.…”

Section: Lorentzian Cross Curvature Solitons On Lorentzian 3-dimensio...mentioning

confidence: 95%

“…Also, other geometric solitons have been studied on locally homogeneous manifolds. For instance, it has been proven that Lie groups with a left-invariant Riemannian metric of dimension of four at most lack non-trivial homogeneous invariant Ricci solitons (see [11][12][13][14]), but there are three-dimensional Riemannian homogeneous Ricci solitons [15,16]. Lauret's work established that every algebraic Ricci soliton on a Lie group with left-invariant Riemannian metric is a homogeneous Ricci soliton [17], and Onda later extended this finding to the case of Lie groups with pseudo-Riemannian left-invariant metric [18].…”

Section: Introductionmentioning

confidence: 99%

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Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

Azami,

Jafari,

Haseeb

et al. 2024

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Section: Lorentzian Cross Curvature Solitons On Lorentzian 3-dimensio...mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

Azami,

Jafari,

Haseeb

et al. 2024

Axioms

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“…Now, we assume that X 3 ≠ 0 and c = 0 . Then the first equation of the system (22) implies that X 3 = − a and the system (22) reduces to Hence, the cases ( 12) and ( 13) are true. ◻ Remark The cases (3)-( 6) of the Theorem 3.6 imply that a non-reductive fourdimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra 1 has non-trivial Killing vector fields.…”

Section: Proof By Definition Of X ♭ We Getmentioning

confidence: 98%

“…In this case, the system (20) reduces to If = 0 then the case (7) holds. If ≠ 0 then the first and the second equations of the system (22) imply that cX 3 = 0 . We consider X 3 = 0 , then The first and fourth equations of the system (23) imply that X 1 = 0 or X 1 = d 2 a 2 .…”

Section: Proof By Definition Of X ♭ We Getmentioning

confidence: 99%

“…In this regard, several results have been derived. For instance, there are no nontrivial homogeneous invariant Ricci solitons on Lie groups with leftinvariant Riemannian metric of dimension at most four (see [14,18,22,29]) and there exists three-dimensional Riemannian homogenous Ricci solitons [4,24]. On a Lie group with left-invariant Riemannian metric, Lauret [25] showed that any algebraic Ricci soliton is a homogenous Ricci soliton.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Ricci Solitons on Non-reductive Four-Dimensional Homogeneous Spaces

Azami

Fasihi-Ramandi

Pirhadi

2023

J Nonlinear Math Phys

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In the present paper, we consider the non-reductive four-dimensional homogeneous spaces and we classify homogeneous generalized Ricci solitons on these spaces. We show that any non-reductive four-dimensional homogeneous space admits the least in a generalized Ricci soliton. Also, we will prove that non-reductive four-dimensional homogeneous spaces have non-trivial Killing vector fields and these spaces exclusive of types A1, A4 and B2 are Einstein manifold and admit in non-trivial homogeneous Ricci solitons.

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On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

Vylegzhanin¹,

Клепиков²,

Родионов³

et al. 2021

Известия АлтГУ

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Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

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Homogeneous Invariant Ricci Solitons on Four-dimensional Lie Groups

Cited by 6 publications

References 0 publications

Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

Generalized Ricci Solitons on Non-reductive Four-Dimensional Homogeneous Spaces

On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

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