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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
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