scholarly journals Gradient Ricci solitons with a conformal vector field

2018 ◽  
Vol 109 (2) ◽  
Author(s):  
Ramesh Sharma
Keyword(s):  
Vector Field ◽  
Ricci Solitons ◽  
Conformal Vector Field ◽  
Gradient Ricci Solitons ◽  
Conformal Vector

scholarly journals Certain results on N(k)-contact metric manifolds

2018 ◽  
Vol 49 (3) ◽  
pp. 205-220
Author(s):  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Characteristic Vector ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons ◽  
Contact Metric Manifolds ◽  
Nullity Distribution ◽  
Characteristic Vector Field

In the present paper we study contact metric manifolds whose characteristic vector field $\xi$ belonging to the $k$-nullity distribution. First we consider concircularly pseudosymmetric $N(k)$-contact metric manifolds of dimension $(2n+1)$. Beside these, we consider Ricci solitons and gradient Ricci solitons on three dimensional $N(k)$-contact metric manifolds. As a consequence we obtain several results. Finally, an example is given.

scholarly journals Closed conformal vector fields on pseudo-Riemannian manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
D. A. Catalano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Geometric Proof ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Local Coordinates

We give here a geometric proof of the existence of certain local coordinates on a pseudo-Riemannian manifold admitting a closed conformal vector field.

scholarly journals On the zeros of a conformal vector field

1974 ◽  
Vol 55 ◽  
pp. 1-3 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Short Note ◽  
Killing Vector ◽  
Connected Components ◽  
Conformal Vector Field ◽  
Totally Geodesic Submanifolds ◽  
Conformal Vector ◽  
Conformal Vector Fields

In [1] S. Kobayashi showed that the connected components of the set of zeros of a Killing vector field on a Riemannian manifold (Mn,g) are totally geodesic submanifolds of (Mn,g) of even codimension including the case of isolated singular points. The purpose of this short note is to give a simple proof of the corresponding result for conformal vector fields on compact Riemannian manifolds. In particular we prove the following

Conformal vector fields on almost co-Kähler manifolds

2021 ◽  
Vol 71 (6) ◽  
pp. 1545-1552
Author(s):  
Uday Chand De ◽  
Young Jin Suh ◽  
Sudhakar K. Chaubey
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Killing Vector Field ◽  
Kahler Manifolds ◽  
Conformal Vector Field ◽  
Kahler Manifold ◽  
Conformal Vector ◽  
Conformal Vector Fields

Abstract In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.

scholarly journals Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 863
Author(s):  
Amira Ishan ◽  
Sharief Deshmukh ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Vector Field ◽  
Nontrivial Solution ◽  
Vector Fields ◽  
Dimensional Sphere ◽  
Conformal Vector Field ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
De Rham

We study the effect of a nontrivial conformal vector field on the geometry of compact Riemannian spaces. We find two new characterizations of the m-dimensional sphere Sm(c) of constant curvature c. The first characterization uses the well known de-Rham Laplace operator, while the second uses a nontrivial solution of the famous Fischer–Marsden differential equation.

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.

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