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The Riemann curvature tensor R(X,Y)Z measures the failure of covariant differentiation to commute: R(X,Y)Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z - ∇_{[X,Y]} Z. Equivalently, it measures the rotation a vector undergoes when parallel-transported around an infinitesimal loop. The curvature tensor is the fundamental invariant of a connection — it vanishes if and only if the manifold is locally flat (isometric to Euclidean space). All other curvature quantities (Ricci, scalar, sectional) are derived from it.
In Euclidean space, the order of partial differentiation does not matter: ∂²f/∂x∂y = ∂²f/∂y∂x. When you replace partial derivatives with covariant derivatives on a manifold, this commutativity generally fails. The Riemann curvature tensor R precisely measures this failure: for vector fields X, Y, Z, the expression R(X,Y)Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z - ∇_{[X,Y]} Z measures how much "differentiating Z first in the Y then X direction" differs from "first in X then Y." The [X,Y] term corrects for the non-commutativity of X and Y as differential operators, isolating the contribution of the geometry.
The curvature tensor R is a (1,3)-tensor: it takes three vector inputs and returns a vector. In coordinates, R(∂ᵢ, ∂ⱼ)∂ₖ = Rˡₖᵢⱼ ∂ₗ, where Rˡₖᵢⱼ = ∂ᵢΓˡⱼₖ - ∂ⱼΓˡᵢₖ + ΓˡᵢₘΓᵐⱼₖ - ΓˡⱼₘΓᵐᵢₖ. Despite the complicated formula, the key message is: R is built algebraically from the Christoffel symbols and their first derivatives, and it transforms as a tensor. It has extensive symmetries: antisymmetry in the first pair and second pair of lowered indices, pair symmetry Rᵢⱼₖₗ = Rₖₗᵢⱼ, and the first Bianchi identity. These reduce the independent components from n⁴ to n²(n²-1)/12.
Geometrically, R(X,Y)v is the rotation that a vector v acquires when parallel transported around an infinitesimal parallelogram spanned by X and Y. If R = 0, parallel transport is path-independent (locally), and the manifold is flat — isometric to Euclidean space in a neighborhood of each point. If R ≠ 0, different paths between the same endpoints produce different parallel transport maps, and the manifold is genuinely curved. The holonomy group (the group of all parallel transport maps around loops at a point) is generated by the curvature.
The Riemann tensor is the master curvature invariant from which all others derive. Contracting one pair of indices gives the Ricci tensor Rᵢⱼ = Rᵏᵢₖⱼ, which encodes how volumes change along geodesics. Contracting again gives the scalar curvature R = gⁱʲRᵢⱼ, a single number at each point. The sectional curvature K(σ) measures curvature in a 2-plane σ ⊂ TpM. In two dimensions, all these reduce to a single function — the Gaussian curvature. In higher dimensions, they carry progressively more refined information about the geometry.