Physics:Curvature collineation

Short description: Vector field that preserves the Riemann tensor

A curvature collineation (often abbreviated to CC) is vector field which preserves the Riemann tensor in the sense that,

[math]\displaystyle{ \mathcal{L}_X R^a{}_{bcd}=0 }[/math]

where [math]\displaystyle{ R^a{}_{bcd} }[/math] are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by [math]\displaystyle{ CC(M) }[/math] and may be infinite-dimensional. Every affine vector field is a curvature collineation.

See also

• Conformal vector field
• Homothetic vector field
• Killing vector field
• Matter collineation
• Spacetime symmetries