Almost Cosymplectic $$(k,\mu )$$-metrics as $$\eta$$-Ricci Solitons

AbstractIn this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k<0$$ k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k<0$$ k < 0 . In addition, a concrete example is constructed to illustrate the above result.

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.

*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Open Mathematics ◽

10.1515/math-2019-0056 ◽

2019 ◽

Vol 17 (1) ◽

pp. 874-882 ◽

Author(s):

Xinxin Dai ◽

Yan Zhao ◽

Uday Chand De

Keyword(s):

Vector Field ◽

Killing Vector ◽

Ricci Soliton ◽

Contact Transformation ◽

Kenmotsu Manifold ◽

Potential Vector ◽

Kenmotsu Manifolds ◽

Almost Kenmotsu Manifolds ◽

Dimension 2 ◽

Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

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 AND CONTACT METRIC MANIFOLDS

Glasgow Mathematical Journal ◽

10.1017/s0017089512000389 ◽

2012 ◽

Vol 55 (1) ◽

pp. 123-130 ◽

Cited By ~ 5

Author(s):

AMALENDU GHOSH

Keyword(s):

Vector Field ◽

Sectional Curvature ◽

Ricci Soliton ◽

Ricci Solitons ◽

Conformally Flat ◽

Contact Metric Manifold ◽

Potential Vector ◽

Contact Metric Manifolds ◽

AbstractWe study on a contact metric manifold M2n+1(ϕ, ξ, η, g) such that g is a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions: (i) M is of pointwise constant ξ-sectional curvature, (ii) M is conformally flat.

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 solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0321 ◽

2019 ◽

Vol 69 (6) ◽

pp. 1447-1458 ◽

Author(s):

Venkatesha ◽

Devaraja Mallesha Naik ◽

H. Aruna Kumara

Keyword(s):

Vector Field ◽

Ricci Soliton ◽

Characteristic Vector ◽

Ricci Solitons ◽

Potential Vector ◽

Open Set ◽

Kenmotsu Manifolds ◽

Almost Ricci Soliton ◽

Characteristic Vector Field ◽

Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

A Note on Ricci Solitons

Symmetry ◽

10.3390/sym12020289 ◽

2020 ◽

Vol 12 (2) ◽

pp. 289 ◽

Author(s):

Sharief Deshmukh ◽

Hana Alsodais

Keyword(s):

Vector Field ◽

Potential Function ◽

Poisson Equation ◽

Energy Function ◽

Ricci Soliton ◽

Ricci Solitons ◽

Potential Vector ◽

In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.

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.

A Note on Solitons with Generalized Geodesic Vector Field

Symmetry ◽

10.3390/sym13071104 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1104

Author(s):

Adara M. Blaga ◽

Amira Ishan ◽

Sharief Deshmukh

Keyword(s):

Vector Field ◽

Killing Vector ◽

Ricci Soliton ◽

Ricci Solitons ◽

General Notion ◽

Killing Vector Field ◽

Potential Vector ◽

Geodesic Vector ◽

Almost Ricci Soliton ◽

We consider a general notion of an almost Ricci soliton and establish some curvature properties for the case in which the potential vector field of the soliton is a generalized geodesic or a 2-Killing vector field. In this vein, we characterize trivial generalized Ricci solitons.