# Curvature collineation

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Vector field that preserves the Riemann tensor

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A **curvature collineation** (often abbreviated to **CC**) is [vector field](/source/Vector_field) which preserves the [Riemann tensor](/source/Riemann_tensor) in the sense that,

- L X R a b c d = 0 {\displaystyle {\mathcal {L}}_{X}R^{a}{}_{bcd}=0}

where R a b c d {\displaystyle R^{a}{}_{bcd}} are the components of the Riemann tensor. The [set](/source/Set_(mathematics)) of all [smooth](/source/Smooth_function) curvature collineations forms a [Lie algebra](/source/Lie_algebra) under the [Lie bracket](/source/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 C C ( M ) {\displaystyle CC(M)} and may be [infinite](/source/Infinity)-[dimensional](/source/Dimension). Every [affine vector field](/source/Affine_vector_field) is a curvature collineation.

## See also

- [Conformal vector field](/source/Conformal_vector_field)

- [Homothetic vector field](/source/Homothetic_vector_field)

- [Killing vector field](/source/Killing_vector_field)

- [Matter collineation](/source/Matter_collineation)

- [Spacetime symmetries](/source/Spacetime_symmetries)

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