Riemann curvature tensor
R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}. The (1,3) tensor that fully characterises spacetime curvature; 20 algebraically independent components in 4D. Vanishes if and only if the manifold is flat. Bianchi identity ∇_λ R^ρ_{σμν} + cyclic = 0 underwrites the divergence-free Einstein tensor.
Definition
The Riemann curvature tensor R^ρ_{σμν} is the (1,3) tensor that fully characterises the curvature of a Riemannian or pseudo-Riemannian manifold. Computed from the Christoffel symbols by R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}, it measures the failure of covariant derivatives to commute and equivalently the holonomy of parallel transport around an infinitesimal loop. The Riemann tensor vanishes identically if and only if the manifold is flat — a global statement, even though it is computed pointwise. In a four-dimensional spacetime it has 256 components reduced by symmetries to 20 algebraically independent ones, encoding the full local curvature information that gravity carries.
The Riemann tensor satisfies algebraic symmetries (R_{ρσμν} = R_{μνρσ} = −R_{σρμν} = −R_{ρσνμ}, plus a cyclic identity in the last three indices) and the differential Bianchi identity ∇_λ R^ρ_{σμν} + ∇_μ R^ρ_{σνλ} + ∇_ν R^ρ_{σλμ} = 0. Contracting the Bianchi identity twice yields ∇^μ G_{μν} = 0, the divergence-free property of the Einstein tensor that lets G_{μν} = (8πG/c⁴) T_{μν} couple consistently to a conserved stress-energy tensor — the structural reason general relativity's field equations have exactly the form they do. The Riemann tensor traces down through the Ricci tensor R_{μν} = R^λ_{μλν} and Ricci scalar R = g^{μν} R_{μν}, the inputs to Einstein's equations.