scholarly journals On warped product gradient η-Ricci solitons

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5791-5801 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Warped Product ◽  
Constant Scalar Curvature ◽  
Ricci Soliton ◽  
Product Manifold ◽  
Potential Vector ◽  
Warped Product Manifold ◽  
Gradient Type ◽  
Potential Vector Field

If the potential vector field of an ?-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by f . We give a way to construct a gradient ?-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

scholarly journals A Study of Generalized Projective P − Curvature Tensor on Warped Product Manifolds

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Uday Chand De ◽  
Abdallah Abdelhameed Syied ◽  
Nasser Bin Turki ◽  
Suliman Alsaeed
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Warped Product ◽  
Einstein Manifold ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Divergence Free

The main aim of this study is to investigate the effects of the P − curvature flatness, P − divergence-free characteristic, and P − symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P − curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P − curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P − divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P − symmetric warped product manifold is considered.

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.

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.

scholarly journals A note on warped product almost quasi-Yamabe solitons

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2009-2016 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Base Manifold ◽  
Warped Product Manifold ◽  
Necessary And Sufficient ◽  
Yamabe Solitons ◽  
Yamabe Soliton

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold to induce a gradient almost quasi-Yamabe soliton on the warped product manifold.

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.

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 Gradient almost Ricci solitons on multiply warped product manifolds

2021 ◽  
Vol 13 (2) ◽  
pp. 386-394
Author(s):  
S. Günsen ◽  
L. Onat
Keyword(s):  
Potential Field ◽  
Warped Product ◽  
Einstein Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Product Manifold ◽  
Rigidity Result ◽  
Warped Product Manifold ◽  
Almost Ricci Soliton

In this paper, we investigate multiply warped product manifold \[M =B\times_{b_1} F_1\times_{b_2} F_2\times_{b_3} \ldots \times_{b_m} F_m\] as a gradient almost Ricci soliton. Taking $b_i=b$ for $1\leq i \leq m$ lets us to deduce that potential field depends on $B$. With this idea we also get a rigidity result and show that base is a generalized quasi-Einstein manifold if $\nabla b$ is conformal.

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.

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