Riemann curvature tensor
/ REE-mahn /
How do you tell, from inside a space, whether it is genuinely curved or merely described in bent coordinates? A non-trivial metric or nonzero Christoffel symbols are not proof — flat space in polar coordinates has both. The Riemann curvature tensor is the honest, coordinate-proof answer: a single tensor that is zero if and only if the space is truly flat, and whose nonzero components measure exactly how much, and in which planes, the space curves.
Its cleanest meaning is operational. Take a vector and parallel-transport it around a tiny closed loop spanned by two directions; in flat space it comes back unchanged, but in curved space it comes back slightly rotated. The Riemann tensor is the machine that turns that loop and that vector into the rotation: the change in the vector equals (a Riemann-tensor contraction) times the area of the loop, to leading order. Equivalently, it measures the failure of two covariant derivatives to commute: (nabla_i nabla_j - nabla_j nabla_i) V^k = R^k_{lij} V^l. In components it is built from the Christoffel symbols and their first derivatives, R^l_{ijk} = partial_j Gamma^l_{ik} - partial_k Gamma^l_{ij} + Gamma^l_{jm} Gamma^m_{ik} - Gamma^l_{km} Gamma^m_{ij}. With four indices it carries a lot of data — in n dimensions it has n^2(n^2-1)/12 independent components after its symmetries (1 in 2-D, 6 in 3-D, 20 in 4-D spacetime). Contracting it gives the Ricci tensor and then the scalar curvature, the pieces that enter Einstein's field equations.
The Riemann tensor is the centerpiece of differential geometry and the mathematical core of general relativity. In two dimensions its single independent component is essentially the Gaussian curvature, tying this advanced machinery back to the curvature of ordinary surfaces. In four-dimensional spacetime it encodes the full tidal gravitational field: the relative acceleration of nearby free-falling particles (geodesic deviation) is governed directly by Riemann, which is why tidal stretching near a black hole is a curvature effect, not a force. The crucial honest point that sets it apart from Christoffel symbols: the Riemann tensor IS a genuine tensor, so it cannot be transformed away — if it is nonzero in one coordinate system it is nonzero in all of them, which is exactly why it, and not the connection, is the true and unambiguous measure of curvature.
On a sphere of radius a the Riemann tensor is nonzero and its single independent piece reduces to the Gaussian curvature K = 1/a^2 — which is why a vector parallel-transported around a closed loop on the sphere comes back rotated by an angle equal to (enclosed area) divided by a^2. On a flat plane, or a cylinder, every component of the Riemann tensor is identically zero, confirming both are intrinsically flat.
On a surface the Riemann tensor collapses to the Gaussian curvature; it vanishes exactly on intrinsically flat spaces.
The Riemann tensor vanishing everywhere is the rigorous definition of a flat space — and unlike the Christoffel symbols, it is a true tensor, so a single nonzero component in any frame proves curvature that no coordinate change can remove. Sign and index-ordering conventions differ across textbooks, so always check an author's definition before comparing formulas.