the Riemann curvature tensor
/ REE-mahn /
How can you tell, from inside a space, whether it is genuinely curved or just drawn in awkward coordinates? Answer: parallel-transport a vector around a tiny loop and see if it comes back rotated. The Riemann curvature tensor is the exact measure of that rotation per unit area — the precise, coordinate-free obstruction to a space being flat. It is the master object from which sectional, Ricci, and scalar curvature are all derived.
It is defined by R(X, Y) Z = nabla_X nabla_Y Z - nabla_Y nabla_X Z - nabla_([X,Y]) Z, the failure of second covariant derivatives to commute. In coordinates R = R^l_ijk involves derivatives and products of the Christoffel symbols. As a (0,4)-tensor R(X,Y,Z,W) = g(R(X,Y)Z, W) it has rich symmetries: antisymmetry in the first pair, R(X,Y,Z,W) = -R(Y,X,Z,W); antisymmetry in the second pair; pair-symmetry R(X,Y,Z,W) = R(Z,W,X,Y); the first (algebraic) Bianchi identity R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0; and the second (differential) Bianchi identity on its covariant derivative. These symmetries cut its independent components down to n^2(n^2-1)/12 — exactly 1 in dimension 2 (the Gaussian curvature) and 20 in dimension 4.
Significance and honesty. R = 0 everywhere if and only if the manifold is locally isometric to Euclidean space, so it is the complete local invariant of a metric. Two warnings. First, conventions differ: do Carmo, Lee, and Kobayashi-Nomizu disagree on signs and index order, so no curvature formula is THE formula — always state your convention. Second, Riemann curvature is the FULL local information; its traces (Ricci, scalar) throw information away. A flat-Ricci or flat-scalar space need not be flat, so never upgrade a weaker curvature condition to the Riemann tensor.
Parallel-transport a vector around a small geodesic quadrilateral with sides of length epsilon in directions X and Y. To leading order the vector returns rotated by epsilon^2 R(X,Y) acting on it. On a sphere of radius a the returned rotation is positive (vectors swing one way); on a saddle it is negative; on the flat plane it is zero. The tensor R is the precise per-area version of this discrepancy.
Holonomy around an infinitesimal loop equals area times curvature: R is the per-area rotation defect.
Sign and index conventions for R differ across textbooks (do Carmo vs Lee vs Kobayashi-Nomizu), so no single formula is canonical — always state yours. And R is strictly stronger than its traces: vanishing Ricci or scalar curvature does NOT imply the manifold is flat.