Curvature collineation

{{Short description|Vector field that preserves the Riemann tensor}} {{Unreferenced|date=April 2026}} A '''curvature collineation''' (often abbreviated to '''CC''') is [[vector field]] which preserves the [[Riemann tensor]] in the sense that,

:<math>\mathcal{L}_X R^a{}_{bcd}=0</math>

where <math>R^a{}_{bcd}</math> are the components of the Riemann tensor. The [[Set (mathematics)|set]] of all [[smooth function|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>CC(M)</math> and may be [[infinity|infinite]]-[[dimension]]al. Every [[affine vector field]] is a curvature collineation.

==See also==

* [[Conformal vector field]] * [[Homothetic vector field]] * [[Killing vector field]] * [[Matter collineation]] * [[Spacetime symmetries]]

[[Category:Mathematics of general relativity]]