the Riemann curvature tensor
/ REE-mahn /
Here is the object that finally answers, without any dependence on coordinates, the question 'is this space actually curved, or does it just look bent because of the coordinates I chose?' The Riemann curvature tensor is the honest measure of curvature. It captures the rotation a vector suffers when parallel-transported around an infinitesimal loop, and it is nonzero if and only if the geometry is genuinely, irreducibly curved, something no clever change of coordinates can undo.
It is assembled from the Christoffel symbols and their derivatives, R^a_bcd = partial_c Gamma^a_db - partial_d Gamma^a_cb + Gamma^a_ce Gamma^e_db - Gamma^a_de Gamma^e_cb, and its cleanest definition is as the commutator of covariant derivatives: (nabla_c nabla_d - nabla_d nabla_c) V^a = R^a_bcd V^b. Its many symmetries cut the number of independent components in four-dimensional spacetime down to twenty, and its derivatives obey the Bianchi identities, which will turn out to guarantee the conservation of energy and momentum.
The Riemann tensor is where curvature becomes physical. A spacetime is flat exactly when every component of R^a_bcd is zero, and where it is nonzero it drives tidal forces, the stretching and squeezing that geodesic deviation makes visible. The essential caveat is one already met with the Christoffel symbols: nonzero connection coefficients do not imply curvature, but a nonzero Riemann tensor does, unavoidably. It is the tensorial verdict that separates a truly warped spacetime from flat space wearing awkward coordinates.
On a sphere of radius a, every independent component of the Riemann tensor is fixed by the single scale a; the surface is uniformly curved, and no coordinate patch, however clever, can flatten it. A cylinder, by contrast, has zero Riemann tensor: it is intrinsically flat and can be unrolled onto a plane without stretching.
Sphere: curved (Riemann nonzero). Cylinder: intrinsically flat (Riemann zero).
Intrinsic curvature, what the Riemann tensor measures, is not the same as the extrinsic bending you see from outside. A cylinder looks bent to us in three dimensions, yet an ant confined to its surface would find its geometry perfectly Euclidean. Only intrinsic curvature enters general relativity.