scholarly journals ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

2016 ◽  
Vol 222 (1) ◽  
pp. 186-209
Author(s):  
RYOSUKE TAKAHASHI
Keyword(s):  
Vector Field ◽  
Recent Work ◽  
Projective Spaces ◽  
Ricci Soliton ◽  
Complete Intersections ◽  
Necessary Condition ◽  
Ricci Solitons ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Fano Manifold

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

scholarly journals Vector Field Energies and Critical Metrics on Kähler Manifolds

2001 ◽  
Vol 162 ◽  
pp. 41-63 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Bilinear Pairing ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals Certain results on Ricci solitons in α-Kenmotsu manifolds

2018 ◽  
Vol 18 (1) ◽  
pp. 11-15
Author(s):  
Rajesh Kumar ◽  
Ashwamedh Mourya
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
Kenmotsu Manifolds

In this paper, we study some curvature problems of Ricci solitons in α-Kenmotsu manifold. It is shown that a symmetric parallel second order-covariant tensor in a α-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (Lvg + 2S) is parallel where V is a given vector field, then the structure (g, V, λ) yield a Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimentional α-Kenmotsu manifolds are obtained. In the last section, we discuss Ricci soliton for 3-dimentional α-Kenmotsu manifolds.

scholarly journals Ricci Solitons in α-Sasakian Manifolds

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Gurupadavva Ingalahalli ◽  
C. S. Bagewadi
Keyword(s):  
Vector Field ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional

We study Ricci solitons in α-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a α-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if ℒVg+2S is parallel where V is a given vector field, then (g,V,λ) is Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimensional α-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional α-Sasakian manifolds are discussed with an example.

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.

LCS-manifolds and Ricci solitons

2021 ◽  
pp. 2150138
Author(s):  
Absos Ali Shaikh ◽  
Biswa Ranjan Datta ◽  
Akram Ali ◽  
Ali H. Alkhaldi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Perfect Fluid ◽  
Einstein Equation ◽  
Curvature Tensor ◽  
Ricci Soliton ◽  
Scalar Function ◽  
Ricci Solitons ◽  
Ricci Collineation ◽  
Yamabe Soliton

This paper is concerned with the study of [Formula: see text]-manifolds and Ricci solitons. It is shown that in a [Formula: see text]-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an [Formula: see text]-manifold, [Formula: see text] is an irrotational vector field, where [Formula: see text] is a non-zero smooth scalar function. It is proved that in a [Formula: see text]-spacetime with generator vector field [Formula: see text] obeying Einstein equation, [Formula: see text] or [Formula: see text] according to [Formula: see text] or [Formula: see text], where [Formula: see text] is a scalar function and [Formula: see text] is the energy momentum tensor. Also, it is shown that if [Formula: see text] is a non-null spacelike (respectively, timelike) vector field on a [Formula: see text]-spacetime with scalar curvature [Formula: see text] and cosmological constant [Formula: see text], then [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and further [Formula: see text] if and only if [Formula: see text]. The nature of the scalar curvature of an [Formula: see text]-manifold admitting Yamabe soliton is obtained. Also, it is proved that an [Formula: see text]-manifold admitting [Formula: see text]-Ricci soliton is [Formula: see text]-Einstein and its scalar curvature is constant if and only if [Formula: see text] is constant. Further, it is shown that if [Formula: see text] is a scalar function with [Formula: see text] and [Formula: see text] vanishes, then the gradients of [Formula: see text], [Formula: see text], [Formula: see text] are co-directional with the generator [Formula: see text]. In a perfect fluid [Formula: see text]-spacetime admitting [Formula: see text]-Ricci soliton, it is proved that the pressure density [Formula: see text] and energy density [Formula: see text] are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost [Formula: see text]-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid [Formula: see text]-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator [Formula: see text], then the matter content cannot be perfect fluid, and further [Formula: see text] with gravitational constant [Formula: see text] implies that [Formula: see text] is a Killing vector field. Finally, in an [Formula: see text]-manifold, it is proved that if the [Formula: see text]-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the [Formula: see text]-curvature tensor, then it is an [Formula: see text]-Einstein manifold.

Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

2021 ◽  
Author(s):  
Akhilesh Yadav ◽  
Kiran Meena
Keyword(s):  
Vector Field ◽  
Necessary Conditions ◽  
Curvature Vector ◽  
Ricci Soliton ◽  
Necessary Condition ◽  
Geodesic Curve ◽  
Necessary And Sufficient Condition ◽  
Gradient Ricci Soliton ◽  
Riemannian Map ◽  
Necessary And Sufficient

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

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 Solitons in β-Kenmotsu Manifolds

2018 ◽  
Vol 56 (1) ◽  
pp. 149-163
Author(s):  
Rajesh Kumar
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Kenmotsu Manifolds

Abstract The object of the present paper is to study Ricci soliton in β-Kenmotsu manifolds. Here it is proved that a symmetric parallel second order covariant tensor in a β-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (ℒVg +2S)is ∇-parallel where V is a given vector field, then the structure (g, V, λ) yields a Ricci soliton. Further, by virtue of this result, we found the conditions of Ricci soliton in β-Kenmotsu manifold to be shrinking, steady and expending respectively. Next, Ricci soliton for 3-dimensional β-Kenmotsu manifold are discussed with an example.

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.

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