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:

\mathcal{L}_X g=\varphi g

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

Affine vector field
Curvature collineation
Homothetic vector field
Killing vector field
Matter collineation
Spacetime symmetries

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