scholarly journals Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ali H. Alkhaldi ◽  
Mohd Danish Siddiqi ◽  
Meraj Ali Khan ◽  
Lamia Saeed Alqahtani
Keyword(s):  
Vector Field ◽  
Potential Function ◽  
Poisson Equation ◽  
Sufficient Conditions ◽  
Potential Vector ◽  
Imperfect Fluid ◽  
Gradient Type ◽  
Necessary And Sufficient ◽  
Yamabe Soliton ◽  
Potential Vector Field

In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field ξ . Furthermore, if the potential vector field ξ of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of η -Ricci-Yamabe soliton on an imperfect fluid GRW spacetime with a harmonic potential function ψ . Finally, we examine necessary and sufficient conditions for a 1 -form η , which is the g -dual of the vector field ξ on imperfect fluid GRW spacetime to be a solution of the Schrödinger-Ricci equation.

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.

Almost quasi-Yamabe solitons on almost cosymplectic manifolds

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 ◽  
Potential Vector Field

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.

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 Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2189
Author(s):  
Pengfei Zhang ◽  
Yanlin Li ◽  
Soumendu Roy ◽  
Santu Dey
Keyword(s):  
Conformal Killing ◽  
Soliton Equation ◽  
Potential Vector ◽  
Research Article ◽  
Cosymplectic Manifolds ◽  
Metric Connection ◽  
Gradient Type ◽  
Yamabe Solitons ◽  
Yamabe Soliton ◽  
Potential Vector Field

The outline of this research article is to initiate the development of a ∗-conformal η-Ricci–Yamabe soliton in α-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of α-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal η-Ricci–Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional α-cosymplectic metric as a ∗-conformal η-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

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 Yamabe Solitons with potential vector field as torse forming

Cubo (Temuco) ◽  
2018 ◽  
Vol 20 (3) ◽  
pp. 37-47
Author(s):  
Yadab ChandraMandal ◽  
Shyamal Kumar Hui
Keyword(s):  
Vector Field ◽  
Potential Vector ◽  
Yamabe Solitons ◽  
Potential Vector Field

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 On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

2014 ◽  
Vol 25 (10) ◽  
pp. 1450093
Author(s):  
T. Ito ◽  
B. Scárdua ◽  
Y. Yamagishi
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Finite Number ◽  
Sufficient Conditions ◽  
First Integral ◽  
Holomorphic Vector Field ◽  
Necessary And Sufficient ◽  
Holonomy Map

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ2. As a sample of our results, we prove that N is biholomorphic to ℂ2 if H2(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form [Formula: see text] where λ1/λ2 ∈ ℚ+. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

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