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

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 ◽  
Potential Vector Field

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

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

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 ◽  
Potential Vector Field

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.

Ricci Solitons on Almost Co-Kähler Manifolds

2018 ◽  
Vol 62 (4) ◽  
pp. 912-922 ◽  
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 ◽  
Potential Vector Field

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.

scholarly journals Ricci solitons on almost Kenmotsu 3-manifolds

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 ◽  
Potential Vector Field

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.

Notes on contact Ricci solitons

2010 ◽  
Vol 54 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Jong Taek Cho
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Homogeneous Manifold ◽  
Ricci Solitons ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Potential 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

2012 ◽  
Vol 10 (01) ◽  
pp. 1220022 ◽  
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 ◽  
Potential 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 contact geometry

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 ◽  
Potential Vector Field

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.

CONTACT GEOMETRY AND RICCI SOLITONS

2010 ◽  
Vol 07 (06) ◽  
pp. 951-960 ◽  
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 ◽  
Potential Vector Field

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.

scholarly journals A Note on Solitons with Generalized Geodesic Vector Field

Symmetry ◽  
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 ◽  
Potential Vector Field

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.

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

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

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 ◽  
Potential Vector Field

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.

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