∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

2018 ◽  
Vol 15 (07) ◽  
pp. 1850120 ◽  
Author(s):  
Amalendu Ghosh ◽  
Dhriti Sundar Patra
Keyword(s):  
Unit Sphere ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Contact Manifolds ◽  
Gradient Ricci Soliton ◽  
Frame Work ◽  
Sasakian Metric

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere [Formula: see text]. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian [Formula: see text]-contact manifolds.

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.

Hypersurfaces of Euclidean Space as Gradient Ricci Solitons

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Hana Al-Sodais ◽  
Haila Alodan ◽  
Sharief Deshmukh
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Ricci Soliton ◽  
Necessary And Sufficient Conditions ◽  
Ricci Solitons ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Necessary And Sufficient

Abstract In this paper we obtain some necessary and sufficient conditions for a hypersurface of a Euclidean space to be a gradient Ricci soliton. We also study the geometry of a special type of compact Ricci solitons isometrically immersed into a Euclidean space.

Almost Ricci solitons isometric to spheres

2019 ◽  
Vol 16 (05) ◽  
pp. 1950073 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Lower Bound ◽  
Poisson Equation ◽  
Ricci Curvature ◽  
Unit Sphere ◽  
Potential Field ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Almost Ricci Soliton

We find a characterization of a sphere using a compact gradient almost Ricci soliton and the lower bound on the integral of Ricci curvature in the direction of potential field. Also, we use Poisson equation on a compact gradient almost Ricci soliton to find a characterization of the unit sphere.

scholarly journals Estimates on the non-compact expanding gradient Ricci solitons

2013 ◽  
Vol 21 (3) ◽  
pp. 95-102
Author(s):  
Xiang Gao ◽  
Qiaofang Xing ◽  
Rongrong Cao
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Linear Growth ◽  
Ricci Soliton ◽  
Potential Functions ◽  
Ricci Solitons ◽  
Positive Ricci Curvature ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Normal Geodesic

Abstract In this paper, we deal with the complete non-compact expanding gradient Ricci soliton (Mn,g) with positive Ricci curvature. On the condition that the Ricci curvature is positive and the scalar curvature approaches 0 towards infinity, we derive a useful estimate on the growth of potential functions. Based on this and under the same assumptions, we prove that ∫t0 Rc (γ'(s) , γ' (s))ds and ∫t0 Rc (γ' (,s). v)ds at least have linear growth, where 7(5) is a minimal normal geodesic emanating from the point where R obtains its maximum. Furthermore, some other results on the Ricci curvature are also obtained.

scholarly journals A New Class of Contact Pseudo Framed Manifolds with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
K. L. Duggal
Keyword(s):  
Vector Field ◽  
Real Function ◽  
Tensor Field ◽  
Contact Manifold ◽  
Ricci Soliton ◽  
Contact Manifolds ◽  
Normal Contact ◽  
New Class ◽  
Almost Ricci Soliton ◽  
Characteristic Vector Field

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds M , g , f , λ , ξ by a real tensor field f of type 1,1 , a real function λ such that f 3 = λ 2 f where ξ is its characteristic vector field. We prove in our main Theorem 2 that M admits a closed 2-form Ω if λ is constant. In 1976, Blair proved that the vector field ξ of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, ξ of a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

2021 ◽  
Author(s):  
Arfah Arfah
Keyword(s):  
General Relativity ◽  
Riemannian Space ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Geometric Properties ◽  
Gradient Ricci Soliton ◽  
Group Manifold

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.

scholarly journals The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons

2018 ◽  
Vol 148 (3) ◽  
pp. 461-482 ◽  
Author(s):  
M. Brozos-Vázquez ◽  
E. García-Río ◽  
S. Gavino-Fernández ◽  
P. Gilkey
Keyword(s):  
Ricci Tensor ◽  
Ricci Soliton ◽  
Complete Classification ◽  
Ricci Solitons ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Locally Homogeneous

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show that the soliton is rigid in dimensions 3 and 4. In the steady case we give a complete classification in dimension 3.

Second Variation of the "Total Scalar Curvature" on Contact Manifolds

1995 ◽  
Vol 38 (1) ◽  
pp. 16-22 ◽  
Author(s):  
D. E. Blair ◽  
D. Perrone
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Local Minimum ◽  
Contact Manifold ◽  
Second Variation ◽  
Contact Manifolds ◽  
Total Scalar Curvature ◽  
The Second Variation ◽  
Sasakian Metric ◽  
Characteristic Vector Field

AbstractLet M2n+1 be a compact contact manifold and 𝓐 the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by and showed that the critical points of I(g) on 𝓐 are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of —I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = ∫M R dVg restricted to 𝓐 cannot have a local minimum at any Sasakian metric.

SOME GAP THEOREMS FOR GRADIENT RICCI SOLITONS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250072 ◽  
Author(s):  
MANUEL FERNÁNDEZ-LÓPEZ ◽  
EDUARDO GARCÍA-RÍO
Keyword(s):  
Lower Bounds ◽  
Ricci Tensor ◽  
Sufficient Conditions ◽  
Ricci Soliton ◽  
Upper And Lower Bounds ◽  
Ricci Solitons ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Necessary And Sufficient ◽  
Gap Theorems

Necessary and sufficient conditions for a gradient Ricci soliton to be Einstein are given, showing that they can be expressed in terms of upper and lower bounds on the behavior of the Ricci tensor when evaluated on the gradient of the potential function of the soliton.

scholarly journals $\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

2021 ◽  
pp. 529
Author(s):  
Mohd Danish Siddiqi ◽  
Mehmet Akif Akyol
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Research Article

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

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