the Riemann curvature tensor
/ REE-mahn /
What is the single quantity that tells you, completely and honestly, how curved a space is at a point — not how curved it looks from outside, but how curved it really is to an inhabitant? It is the Riemann curvature tensor. It captures, in one machine, the fact that on a curved space parallel transport around a tiny loop brings vectors home rotated, and equivalently that taking two covariant derivatives in different orders gives different answers. Where it vanishes, the space is flat; where it does not, it measures exactly how the geometry departs from flatness.
Precisely, the Riemann tensor R takes three vector fields and returns a fourth, defined by R(X, Y)Z = nabla_X nabla_Y Z - nabla_Y nabla_X Z - nabla_([X, Y]) Z. Read this as the failure of covariant derivatives to commute: on flat space the order would not matter and R would be zero; on a curved space the mismatch is precisely the rotation a vector picks up going around the infinitesimal loop spanned by X and Y. In coordinates its components R^l_ijk are built from the Christoffel symbols and their first derivatives. It is a genuine tensor, so 'R = 0 at a point' is a coordinate-independent fact — unlike the Christoffel symbols, which can always be zeroed at a point.
The Riemann tensor is the master object of the subject: it is the full intrinsic record of curvature, and everything else (sectional, Ricci, and scalar curvature) is obtained from it by averaging or contracting. Its many symmetries cut its independent components down sharply — in dimension 2 a single number suffices, the Gaussian curvature of the Theorema Egregium. The honest punchline is that Gauss's intrinsic curvature of surfaces is exactly the 2-dimensional case of Riemann's tensor; Riemann's achievement was to find the right object in every dimension and to free curvature, once and for all, from any need for a surrounding space.
On a round sphere of radius a, the Riemann tensor is as simple as possible: every sectional curvature equals 1/a^2, a positive constant. On the flat plane it is identically zero in every coordinate system. The single fact 'R is not zero on the sphere' is what forbids any distortion-free flat map of the Earth — a direct consequence of nonzero curvature.
R vanishes exactly when a space is flat; on the sphere it cannot, which is why no flat map of Earth can be faithful.
The Riemann tensor measures intrinsic curvature — what an inhabitant can detect — and is unrelated to how a space might be bent into a higher dimension. A flat sheet rolled into a cylinder has R = 0: rolling does not change intrinsic curvature, a point students routinely miss.