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A '''conformal vector field''' (often '''conformal Killing vector field''' and occasionally '''conformal''' or '''conformal collineation''') of a [[Riemannian manifold]] <math>(M,g)</math> is a [[vector field]] <math>X</math> that satisfies:

:<math>\mathcal{L}_X g=\varphi g</math>

for some smooth real-valued function <math>\varphi</math> on <math>M</math>, where <math>\mathcal{L}_X g</math> denotes the [[Lie derivative]] of the metric <math>g</math> with respect to <math>X</math>.  In the case that <math>\varphi</math> is identically zero, <math>X</math> is called a [[Killing vector field]].

==See also==

* [[Affine vector field]]
* [[Curvature collineation]]
* [[Homothetic vector field]]
* [[Killing vector field]]
* [[Matter collineation]]
* [[Spacetime symmetries]]

{{relativity-stub}}
[[Category:Mathematical methods in general relativity]]