scholarly journals RICCI SOLITONS ON HOPH HYPERSURFACES IN SASAKIAN SPACE FORM

2017 ◽  
pp. 387
Author(s):  
Zahra Nazari ◽  
Esmail Abedi
Keyword(s):  
Space Form ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Principal Curvatures ◽  
Sasakian Space Form ◽  
Potential Vector

We are studying Ricci solitons on Hoph hypersurfaces in Sasakian space formfM2n+1(c). The rst, we prove that Hoph hypersurfaces of a Sasakian space formfM2n+1(c < 1) with two distinct principal curvatures is shrinking and for c 1,Hoph hypersurfaces with two distinct principal curvatures of a Sasakian space formfM2n+1(c) does not admit a Ricci soliton. We show that there is not any Hoph hyper-surfaces with two distinct principal curvatures in a Sasakian space form fM2n+1(c)with a -Ricci soliton (and Ricci soliton) such that potential vector eld is the Reebvector eld.Then we prove that Hoph hypersurfaces in Sasakian space form fM2n+1(c) withc = 1 does not admit a - Ricci soliton with potential vector eld U and we showthat Ricci soliton on Hoph hypersurfaces M in Sasakian space form fM2n+1(c <

scholarly journals Ricci solitons on QR-hypersurfaces of a quaternionic space form ℚn

2016 ◽  
Vol 8 (2) ◽  
pp. 271-281
Author(s):  
Z. Nazari ◽  
E. Abedi
Keyword(s):  
Space Form ◽  
Shape Operator ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Quaternionic Space Form ◽  
Eigenvalue Zero ◽  
Quaternionic Space

Abstract The purpose of this paper is to study Ricci solitons on QR-hypersurfaces M of a quaternionic space form ℚn such that the shape operator A with respect to N has one eigenvalue. We prove that Ricci soliton on QR-hypersurfaces M with eigenvalue zero is steady and for eigenvalue nonzero is shrinking.

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.

scholarly journals Ricci soliton and η-Ricci soliton on Generalized Sasakian space form

Filomat ◽  
2017 ◽  
Vol 31 (13) ◽  
pp. 4051-4062 ◽  
Author(s):  
Sampa Pahan ◽  
Tamalika Dutta ◽  
Arindam Bhattacharyya
Keyword(s):  
Space Form ◽  
Ricci Soliton ◽  
Sasakian Space Form ◽  
Generalized Sasakian Space Form

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

2021 ◽  
Author(s):  
Wenjie Wang
Keyword(s):  
Vector Field ◽  
Lie Group ◽  
Local Structure ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Contact Transformation ◽  
Potential Vector ◽  
Cosymplectic Manifold ◽  
Almost Cosymplectic Manifold ◽  
Potential Vector Field

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.

scholarly journals RICCI SOLITONS AND CONTACT METRIC MANIFOLDS

2012 ◽  
Vol 55 (1) ◽  
pp. 123-130 ◽  
Author(s):  
AMALENDU GHOSH
Keyword(s):  
Vector Field ◽  
Sectional Curvature ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Conformally Flat ◽  
Contact Metric Manifold ◽  
Potential Vector ◽  
Contact Metric Manifolds ◽  
Potential Vector Field

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

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.

scholarly journals ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

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 ◽  
Potential 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.

scholarly journals A Note on Ricci Solitons

Symmetry ◽  
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 ◽  
Potential Vector Field

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.

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