title: Curvature collineation
text: # 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.
  
  