Curvature collineation

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

${\mathcal {L}}_{X}R^{a}{}_{bcd}=0$

where $R^{a}{}_{bcd}$ 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 $CC(M)$ 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