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