Physics:Conformal vector field

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

[math]\displaystyle{ \mathcal{L}_X g=\varphi g }[/math]

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