A Killing vector field
K is a field along which the Lie derivative of the metric tensor g is zero.
Section 3 is devoted to the study of Riemannian manifolds ([M.sup.n], g) whose metric is Ricci almost soliton with a conformal Killing vector field
. In section 4, we studied Ricci almost solitons on Riemannian manifolds with respect to semi-symmetric metric connection and obtain a necessary and sufficient condition of a Ricci almost soliton on Riemannian manifold with respect to semi-symmetric metric connection to be Ricci almost soliton on Riemannian manifold with respect to Levi-Civita connection.
In particular, a conformal vector field with a vanishing conformal coefficient reduces to a Killing vector field
If V is the gradient of a smooth function f, up to the addition of a Killing vector field
, then we can replace V by [bar.[nabla]f and ([bar.M], [bar.g], [PHI], V) is called a gradient ARS-manifold for which the evolution equation (6) assumes the form
In any almost cosymplectic manifold [[nabla].sub.[xi]][xi] = 0 and [xi] is [nabla]-parallel if and only if it is a Killing vector field
. This occurs in particular in any cosymplectic manifold.
We also consider the skew-symmetric Killing vector field
V defined by
If [xi] is a killing vector field
, then [M.sup.n] is called a K-contact Riemannian manifold (5).
For a stationary space-time metric, there exists a vector field called a time-like Killing vector field
. That means that the space-time metric tensor [g.sub.[mu]v] can be written in a fashion that is independent of the space-time coordinate picked out by the Killing vector field
, and that this vector field is time-like with respect to the metric.
A Riemannian submersion [pi] : N [right arrow] M of a 3-dimensional Riemannian manifold N over a surface M will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field
In particular, an event horizon is called a Killing horizon if it is represented by a null hypersurface which admits a Killing vector field
. Most important family is the Kerr-Newman black holes.
Keywords and Phrases: Quasi Einstein manifolds, Super quasi Einstein manifolds, Generalized quasi-Einstein manifolds, Killing vector field
, Harmonic vector field, Projective Killing vector field
, Conformal Killing vector field
, Sectional curvature.
Note that VV is a Killing vector field
if and only if the fundamental tensor T vanishes identically, that is, the fibers are totally geodesies.