Homogeneous Cotton solitons

Calviño-Louzao, Esteban; Hervella, Luis; Seoane-Bascoy, J.; Vázquez-Lorenzo, Ramón

doi:10.1088/1751-8113/46/28/285204

Cited by 9 publications

(6 citation statements)

References 24 publications

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“…For example, see [16][17][18] for Cotton-like flow in the Horava-Lifshitz gravity context. See also [19][20][21] where Cotton solitons were found. The lay-out of the paper is as follows: in section 2, we linearize the Cotton flow equation about a generic background and identify the symmetries of the equation.…”

Section: Jhep06(2015)136mentioning

confidence: 98%

“…In fact let us give an example in the Riemannian setting with a constant λ [19]. For constant λ, there are many in the Lorentzian setting [19][20][21]. Consider the following metric…”

Section: Jhep06(2015)136mentioning

confidence: 99%

“…Using this fact we can define a new object that we shall call Topologically Massive solitons that solve 20) which is satisfied by (6.14) with the following choices…”

Section: Jhep06(2015)136mentioning

confidence: 99%

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More on Cotton flow

Kilicarslan

Dengiz

Tekin

2015

J. High Energ. Phys.

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Cotton flow tends to evolve a given initial metric on a three manifold to a conformally flat one. Here we expound upon the earlier work on Cotton flow and study the linearized version of it around a generic initial metric by employing a modified form of the DeTurck trick. We show that the flow around the flat space, as a critical point, reduces to an anisotropic generalization of linearized KdV equation with complex dispersion relations one of which is an unstable mode, rendering the flat space unstable under small perturbations. We also show that Einstein spaces and some conformally flat non-Einstein spaces are linearly unstable. We refine the gradient flow formalism and compute the second variation of the entropy and show that generic critical points are extended Cotton solitons. We study some properties of these solutions and find a Topologically Massive soliton that is built from Cotton and Ricci solitons. In the Lorentzian signature, we also show that the pp-wave metrics are both Cotton and Ricci solitons.

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Section: Jhep06(2015)136mentioning

confidence: 98%

“…In fact let us give an example in the Riemannian setting with a constant λ [19]. For constant λ, there are many in the Lorentzian setting [19][20][21]. Consider the following metric…”

Section: Jhep06(2015)136mentioning

confidence: 99%

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More on Cotton flow

Kilicarslan

Dengiz

Tekin

2015

J. High Energ. Phys.

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“…Since x is an eigenvector of Q, by the third term of (7) we have A ¼ B ¼ 0. Next, making use of (17) and Lemma 1, we directly compute the components of C as follows: C 11 ¼ ð' e 2 SÞðe 3 ; e 1 Þ À ð' e 3 SÞðe 2 ; e 1 Þ ¼ ð' fe RicÞðx; eÞ À ð' x RicÞðfe; eÞ ¼ ÀRicð' fe x; eÞ À Ricðx; ' fe eÞ À xðZÞ þ Ricð' x fe; eÞ þ Ricðfe; ' x eÞ…”

Section: Preliminariesmentioning

confidence: 99%

“…Meanwhile, for a non-Riemannian case, they gave the existence of Lorentzian Cotton solitons. Furthermore, E. Calvin ˜o-Louzao et al studied left-invariant Cotton solitons on homogeneous manifolds, see [17].…”

Section: Introductionmentioning

confidence: 99%

Three dimensional contact metric manifolds with Cotton solitons

Chen

2021

Hiroshima Math. J.

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In this article we study a three dimensional contact metric manifold M 3 with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with Qx ¼ rx, where r is a smooth function on M constant along Reeb vector field x and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with x or is a gradient vector field. Moreover, if r is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: SUð2Þ or SOð3Þ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field x. Secondly, it is proved that a ðk; m; nÞ-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that M is Sasakian, flat, a contact metric ð0; À4Þ-space or a contact metric ðk; 0Þ-space with k < 1 and k 0 0. For the potential vector field being orthogonal to x, if n is constant we prove that M is either Sasakian, or a ðk; mÞ-contact metric space.

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Remarks on Cotton solitons

Poddar

2024

Lett Math Phys

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Homogeneous Cotton solitons

Abstract: Left-invariant Cotton solitons on homogeneous manifolds are determined. Moreover, algebraic Cotton solitons are studied providing examples of non-invariant Cotton solitons, both in the Riemannian and Lorentzian homogeneous settings.1991 Mathematics Subject Classification. 53C50, 53B30.

Cited by 9 publications

References 24 publications

More on Cotton flow

More on Cotton flow

Three dimensional contact metric manifolds with Cotton solitons

Remarks on Cotton solitons

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