Ricci solitons on Lorentzian Walker three-manifolds

Calvaruso, Giovanni; Leo, Barbara De

doi:10.1007/s10474-010-0049-z

Cited by 22 publications

(15 citation statements)

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“…A survey and further references on the geometry of Ricci solitons may be found in [10]. First introduced and studied in the Riemannian case, Ricci solitons have been investigated in pseudo-Riemannian settings, with special attention to the Lorentzian case [3,8,11,25]. The Ricci soliton equation also appears to be related to String Theory.…”

Section: Introductionmentioning

confidence: 99%

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Calvaruso

Fino

2012

Can. j. math.

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Section: Introductionmentioning

confidence: 99%

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Calvaruso

Fino

2012

Can. j. math.

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“…In the pseudo-Riemannian case, there are shrinking homogeneous Ricci solitons discovered in [18], while all threedimensional homogeneous Lorentzian Ricci solitons are classified in [3]. Other results about Lorentzian Ricci solitons are found in [2], [4], [6].…”

Section: Introductionmentioning

confidence: 94%

Examples of Algebraic Ricci Solitons in the Pseudo-Riemannian Case

Onda

2014

Acta Math. Hungar.

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This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper, we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.of the central problems in differential geometry. For instance, Perelman [19] was able to prove Thurston's geometrization conjecture.From the definition of the Ricci flow, a fixed point of the Ricci flow is a Ricci-flat metric, and a fixed point of the normalized Ricci flow is an Einstein metric. From Proposition 2.3, Ricci solitons change only by diffeomorphism and rescaling, and are regarded as generalized fixed points. In other words, let M(M n ) be the space of Riemannian metrics on M n , and D(M n ) the diffeomorphism group of M n . Considering the dynamical system of the Ricci flow on the moduli space M(M n )/D(M n ) × R + , we regard Ricci solitons as fixed points.In this paper, we study left-invariant pseudo-Riemannian metrics and algebraic Ricci solitons on Loretzian Lie groups. The concept of an algebraic Ricci soliton was first introduced by Lauret in the Riemannian case (see [13]). Lauret proved that algebraic Ricci solitons on homogeneous Riemannian manifolds are Ricci solitons. In general, problems for Ricci solitons are second-order differential equations. However, problems for algebraic Ricci solitons are algebraic equations. Therefore, algebraic Ricci solitons allow us to construct Ricci solitons in an algebraic way, i.e., using algebraic Ricci soliton theory, the study of Ricci solitons on homogeneous manifolds becomes algebraic. So far, Ricci solitons have been studied in the Riemannian case. Recently, the study of Ricci solitons in the pseudo-Riemannian setting has started with special attention to the Lorentzian case. In the Riemannian case, all homogeneous nontrivial Ricci solitons are expanding Ricci solitons. In the pseudo-Riemannian case, there are shrinking homogeneous non-trivial Ricci solitons discovered in [18], while all vector fields of these Ricci solitons are not left-invariant. And 3-dimensional homogeneous Lorentzian Ricci solitons with left-invariant vector fields are classified in [3]. Other results about Lorentzian Ricci solitons are found in [2], [4], [6], [7].In this paper, we study algebraic Ricci solitons on Lorentzian Lie groups in the Lorentzian case. By using algebraic Ricci soliton theory, we can construct homogeneous Lorentzian Ricci solitons in an algebraic way. For example, we can construct the Ricci soliton in [18] by using algebraic Ricci solitons and Theorem 2.5. Recall that all homogeneous non-trivial solvsolitons are expanding in the Riemannian setting. In this paper, we construct Lorentzian algebraic Ricci solitons on the Heisenberg group H N and the oscillator groups G m (λ) and on three-dimensional Lorentzian Lie groups. In particular, we obtain new Lorentzian Ricci solitons on H N and G m (λ). This paper is organized as follows. In Sec...

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“…Thus, by using (2), (6) and (8) in (1), a standard calculation gives that the three-dimensional Lie group H 2 × R, g is a Ricci soliton if and only if the following system holds,…”

Section: Ricci Solitons Of 3-dimensional H 2 × R Lie Groupmentioning

confidence: 99%

“…Lorentzian Ricci solitons have been intensively studied, showing many essential differences with respect to the Riemannian case (see [2], [4], [3], [8], [23], [22]). In fact, although there exist three-dimensional Riemannian homogeneous Ricci solitons [1], [20], there are no left-invariant Riemannian Ricci solitons on three-dimensional Lie groups [14] (see also [18] and [24]).…”

Section: Introductionmentioning

confidence: 99%

Ricci solitons of the <inline-formula><tex-math id="M1">$ \mathbb{H}^{2} \times \mathbb{R} $</tex-math></inline-formula> Lie group

Belarbi¹

2020

era

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Ricci solitons on Lorentzian Walker three-manifolds

Abstract: We investigate Ricci solitons on Lorentzian three-manifolds (M, g f ) admitting a parallel degenerate line field. For several classes of these manifolds, described in terms of the defining function f , the existence of nontrivial Ricci solitons is proved.

Cited by 22 publications

References 9 publications

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Examples of Algebraic Ricci Solitons in the Pseudo-Riemannian Case

Ricci solitons of the <inline-formula><tex-math id="M1">$ \mathbb{H}^{2} \times \mathbb{R} $</tex-math></inline-formula> Lie group

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