The covariant derivative of the Riemann tensor is given by R_(λμνκ;η) = 1/2 d/(dx^η)((d^2 g_λν)/(dx^κ dx^μ) - (d^2 g_μν)/(dx^κ dx^λ) - (d^2 g_λκ)/(dx^μ dx^ν) + (d^2 g_μκ)/(dx^ν dx^λ)). Permuting ν, κ, and η gives the Bianchi identities R_(λμνκ;η) + R_(λμην;κ) + R_(λμκη;ν) = 0, which can be written concisely as R^α_(β[λμ;ν]) = 0 , where T_[a_1 ...a_n] denoted the antisymmetric tensor part.

contracted Bianchi identities | Einstein field equations | Ricci curvature tensor | Riemann tensor