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A curvature collineation is vector field which preserves the Riemann tensor in the sense that,

where R a b c d {\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. The Lie algebra is denoted by C C {\displaystyle CC} and may be infinite-dimensional. Every affine vector field is a curvature collineation.

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