Lie derivatives are applied to conformal invariance, first integrals, Killing vector
fields, and symmetries of differential equations.
In particular, a conformal vector field with a vanishing conformal coefficient reduces to a Killing vector
An ACV reduces to a conformal Killing vector
, briefly denoted by CKV, if K is proportional to [bar.
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
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.
xi]][xi] = 0 and [xi] is [nabla]-parallel if and only if it is a Killing vector
We also consider the skew-symmetric Killing vector
field V defined by
For a stationary space-time metric, there exists a vector field called a time-like Killing vector
u] is conformal Killing vector
([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) on each [H.
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.
Moreover, [xi] is a Killing vector
field if and only if M is a quasi-Sasakian manifold, that is d[PHI] = 0, where [PHI](X, Y) = g(X, fY).
Among different spacetime symmetries isometries or the Killing vectors
(KVs) have the importance that they help in understanding the geometric properties of spaces also corresponding to each isometries there is some conserved quantity.