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:

${\mathcal {L}}_{X}g=\varphi g$

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