Curvature collineation

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

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

where ${\displaystyle R^{a}{}_{bcd}}$ 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 ${\displaystyle CC(M)}$ and may be infinite-dimensional. Every affine vector field is a curvature collineation.