The k-almost Yamabe solitons and almost coKähler manifolds

In this paper, we study almost coKähler manifolds admitting [Formula: see text]-almost Yamabe solitons [Formula: see text]. First, we obtain a classification of almost coKähler [Formula: see text]-manifolds admitting nontrivial closed [Formula: see text]-almost Yamabe solitons. Next, we consider an almost [Formula: see text]-coKähler manifold admitting a nontrivial [Formula: see text]-almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector field. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector field, we also obtain two results.

Almost quasi-Yamabe solitons on almost cosymplectic manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988782050070x ◽

2020 ◽

Vol 17 (05) ◽

pp. 2050070

Author(s):

Xiaomin Chen

Keyword(s):

Vector Field ◽

Riemannian Product ◽

Reeb Vector ◽

Potential Vector ◽

Cosymplectic Manifolds ◽

Almost Kähler Manifold ◽

Almost Kähler ◽

Yamabe Solitons ◽

Yamabe Soliton ◽

In this paper, we study almost cosymplectic manifolds admitting almost quasi-Yamabe solitons [Formula: see text]. First, we prove that an almost cosymplectic [Formula: see text]-manifold is locally isomorphic to a Lie group if [Formula: see text] is a nontrivial closed quasi-Yamabe soliton. Next, we consider an almost [Formula: see text]-cosymplectic manifold admitting a nontrivial almost quasi-Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector filed. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector filed, we also obtain two results. Finally, for a closed almost quasi-Yamabe soliton on compact [Formula: see text]-cosymplectic manifolds, we prove that it is trivial if [Formula: see text] is nonnegative, where [Formula: see text] is the scalar curvature.

Ricci almost solitons and contact geometry

Advances in Geometry ◽

10.1515/advgeom-2019-0026 ◽

2019 ◽

Vol 0 (0) ◽

Author(s):

Amalendu Ghosh

Keyword(s):

Vector Field ◽

Contact Manifold ◽

Ricci Soliton ◽

Jacobi Field ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Rigidity Results ◽

Parallel Ricci Tensor ◽

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

Ricci Solitons on Almost Co-Kähler Manifolds

Canadian Mathematical Bulletin ◽

10.4153/s0008439518000632 ◽

2018 ◽

Vol 62 (4) ◽

pp. 912-922 ◽

Cited By ~ 3

Author(s):

Yaning Wang

Keyword(s):

Vector Field ◽

Ricci Soliton ◽

Ricci Solitons ◽

Constant Length ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Constant Coefficients ◽

Rigid Motions ◽

AbstractIn this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$-Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.

Ricci solitons on almost Kenmotsu 3-manifolds

Open Mathematics ◽

10.1515/math-2017-0103 ◽

2017 ◽

Vol 15 (1) ◽

pp. 1236-1243 ◽

Author(s):

Yaning Wang

Keyword(s):

Vector Field ◽

Hyperbolic Space ◽

Lie Group ◽

Ricci Soliton ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Gradient Ricci Soliton ◽

Abstract Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.

CONTACT GEOMETRY AND RICCI SOLITONS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887810004646 ◽

2010 ◽

Vol 07 (06) ◽

pp. 951-960 ◽

Cited By ~ 18

Author(s):

JONG TAEK CHO ◽

RAMESH SHARMA

Keyword(s):

Vector Field ◽

Vector Fields ◽

Ricci Soliton ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Base Manifold ◽

Gradient Ricci Soliton ◽

Unit Tangent Bundle ◽

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

Notes on contact Ricci solitons

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000571 ◽

2010 ◽

Vol 54 (1) ◽

pp. 47-53 ◽

Cited By ~ 12

Author(s):

Jong Taek Cho

Keyword(s):

Vector Field ◽

Ricci Soliton ◽

Homogeneous Manifold ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

AbstractA compact contact Ricci soliton (whose potential vector field is the Reeb vector field) is Sasaki–Einstein. A compact contact homogeneous manifold with a Ricci soliton is Sasaki–Einstein.

ALMOST CONTACT 3-MANIFOLDS AND RICCI SOLITONS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812200228 ◽

2012 ◽

Vol 10 (01) ◽

pp. 1220022 ◽

Cited By ~ 8

Author(s):

JONG TAEK CHO

Keyword(s):

Vector Field ◽

Sectional Curvature ◽

Ricci Soliton ◽

Constant Sectional Curvature ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Transversal Vector Field ◽

A Kenmotsu 3-manifold M admitting a Ricci soliton (g, w) with a transversal potential vector field w (orthogonal to the Reeb vector field) is of constant sectional curvature -1. A cosymplectic 3-manifold admitting a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.

Ricci almost solitons and gradient Ricci almost solitons in $(k,\mu)$-paracontact geometry

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v37i3.33027 ◽

2017 ◽

Vol 37 (3) ◽

pp. 119 ◽

Author(s):

Uday Chand De ◽

Krishanu Mandal

Keyword(s):

Vector Field ◽

Constant Curvature ◽

Einstein Manifold ◽

Dimensional Manifold ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Negative Constant ◽

The purpose of this paper is to study Ricci almost soliton and gradient Ricci almost soliton in $(k,\mu)$-paracontact metric manifolds. We prove the non-existence of Ricci almost soliton in a $(k,\mu)$-paracontact metric manifold $M$ with $k<-1$ or $k>-1$ and whose potential vector field is the Reeb vector field $\xi$. Further, if the metric $g$ of a $(k,\mu)$-paracontact metric manifold $M^{2n+1}$ with $k\neq-1$ is a gradient Ricci almost soliton, then we prove either the manifold is locally isometric to a product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative constant curvature equal to $-4$, or, $M^{2n+1}$ is an Einstein manifold.

Almost $$*$$-Ricci soliton on paraKenmotsu manifolds

Arabian Journal of Mathematics ◽

10.1007/s40065-019-00269-7 ◽

2019 ◽

Vol 9 (3) ◽

pp. 715-726 ◽

Author(s):

V. Venkatesha ◽

H. Aruna Kumara ◽

Devaraja Mallesha Naik

Keyword(s):

Vector Field ◽

Constant Curvature ◽

Three Dimensional ◽

Ricci Soliton ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Almost Ricci Soliton ◽

Abstract We consider almost $$*$$ ∗ -Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of $$\eta $$ η -Einstein paraKenmotsu manifold is $$*$$ ∗ Ricci soliton, then M is Einstein. Next, we show that if $$\eta $$ η -Einstein paraKenmotsu manifold admits a gradient almost $$*$$ ∗ -Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field $$\xi $$ ξ . Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature $$-1$$ - 1 . An illustrative example is given to support the obtained results.

N(k)-paracontact three metric as a Eta-Ricci soliton

SERIES III - MATEMATICS, INFORMATICS, PHYSICS ◽

10.31926/but.mif.2020.13.62.2.16 ◽

2021 ◽

Vol 13(62) (2) ◽

pp. 581-594

Author(s):

Debabrata Kar ◽

Pradip Majhi

Keyword(s):

Vector Field ◽

Killing Vector ◽

Three Dimensional ◽

Ricci Soliton ◽

Ricci Solitons ◽

Reeb Vector ◽

Potential Vector ◽

Reeb Vector Field ◽

Infinitesimal Automorphism ◽

In this paper, we study Eta-Ricci soliton (η-Ricci soliton) on three dimensional N(k)-paracontact metric manifolds. We prove that the scalar curvature of an N(k)-paracontact metric manifold admitting η-Ricci solitons is constant and the manifold is of constant curvature k. Also, we prove that such manifolds are Einstein. Moreover, we show the condition of that the η-Ricci soliton to be expanding, steady or shrinking. In such a case we prove that the potential vector field is Killing vector field. Also, we show that the potential vector field is an infinitesimal automorphism or it leaves the structure tensor in the direction perpendicular to the Reeb vector field ξ. Finally, we illustrate an example of a three dimensional N(k)-paracontact metric manifold admitting an η-Ricci soliton