Conformal vector field

A conformal vector field (often conformal Killing vector field and occasionally conformal or conformal collineation) of a Riemannian manifold ( M , g ) is a vector field X that satisfies:

for some smooth real-valued function φ on M , where L X g denotes the Lie derivative of the metric g with respect to X . In the case that φ is identically zero, X is called a Killing vector field.