# Killing vector

## Killing vector

[¦kil·iŋ ¦vek·tər]
(mathematics)
An element of a vector field in a geometry that generates an isometry.
References in periodicals archive ?
Lie derivatives are applied to conformal invariance, first integrals, Killing vector fields, and symmetries of differential equations.
The point is that, for (4), R = 2m is also a Killing horizon, implying that the timelike Killing vector [[chi].sub.(0)] becomes null on that surface and, still worse, becomes spacelike, inside it.
It is the Lie algebra of continuously differentiable transformations K = [K.sup.a]([partial derivative]/[partial derivative][x.sup.a]), where [K.sup.a] = [K.sup.a] ([x.sup.b]) are the components of the vector field K, known as a Killing vector (KV) field.
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 particular, a conformal vector field with a vanishing conformal coefficient reduces to a Killing vector field.
An ACV reduces to a conformal Killing vector, briefly denoted by CKV, if K is proportional to [bar.g].
It admits a five dimensional Lie algebra of Killing vector fields generated by a time translation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], two spatial translations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] plus two further Killing vector fields:
A conformal Killing vector X is defined by the action of [[Laplace].sub.X] on the metric tensor field g so that
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.
If [xi] is a killing vector field, then [M.sup.n] is called a K-contact Riemannian manifold (5).
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