In general relativity and tensor calculus, the Palatini identity is
where denotes the variation of Christoffel symbols and indicates covariant differentiation.
The "same" identity holds for the Lie derivative . In fact, one has
where denotes any vector field on the spacetime manifold .
The Riemann curvature tensor is defined in terms of the Levi-Civita connection as
Its variation is
While the connection is not a tensor, the difference between two connections is, so we can take its covariant derivative
Solving this equation for and substituting the result in , all the -like terms cancel, leaving only
Finally, the variation of the Ricci curvature tensor follows by contracting two indices, proving the identity