Palatini identity

In general relativity and tensor calculus, the Palatini identity is:

where ${\displaystyle \delta \Gamma _{\nu \sigma }^{\rho }}$ denotes the variation of Christoffel symbols and ${\displaystyle \nabla _{\rho }}$ indicates covariant differentiation.

Proof can be found in the entry Einstein–Hilbert action.

The "same" identity holds for the Lie derivative ${\displaystyle {\mathcal {L}}_{\xi }R_{\sigma \nu }}$. In fact, one has:

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