Curved spaces have a property that flat spaces don’t: if you carry a vector around a closed loop, it comes back rotated. The amount of rotation depends on the path and the curvature. The Riemann curvature tensor captures this precisely — it’s the mathematical object that says “how much” and “in what way” spacetime is curved at each point.

Parallel transport reveals curvature

On a flat plane, you can slide a vector from one point to another and it stays “the same” — same direction, same length. On a curved surface, there’s no unique way to do this. The best you can do is parallel transport: move the vector along a curve, keeping it “as straight as possible” at each step.

On a flat surface, parallel transport around a closed loop returns the vector unchanged. On a curved surface, the vector comes back rotated. The angle deficit is proportional to the curvature enclosed by the loop.

The Riemann tensor

The Riemann curvature tensor $R^\rho{}_{\sigma\mu\nu}$ is a rank-(1,3) tensor defined in terms of the Christoffel symbols and their derivatives:

$$R^\rho{}{\sigma\mu\nu} = \partial\mu \Gamma^\rho{}{\nu\sigma} - \partial\nu \Gamma^\rho{}{\mu\sigma} + \Gamma^\rho{}{\mu\lambda}\Gamma^\lambda{}{\nu\sigma} - \Gamma^\rho{}{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$$

In $n$ dimensions, this tensor has $n^2(n^2-1)/12$ independent components. In 4D spacetime: 20 independent components. These 20 numbers at each point completely characterise the curvature.

Physically, $R^\rho{}_{\sigma\mu\nu}$ measures geodesic deviation: the relative acceleration between two neighbouring freely falling particles. This is the tidal force — what stretches you as you fall into a black hole.

Contractions: Ricci tensor and scalar

The Riemann tensor is too complex to put directly into a field equation (20 components, but the metric has only 10). We need contractions:

Ricci tensor — contract the first and third indices:

$$R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu}$$

This is a symmetric (0,2) tensor with 10 independent components — the same as the metric. It measures how volumes change as they’re parallel-transported. Positive $R_{\mu\nu}$ means volumes shrink (gravity focuses geodesics); negative means they expand.

Ricci scalar — contract again:

$$R = g^{\mu\nu} R_{\mu\nu}$$

A single number at each point: the total scalar curvature. On a sphere of radius $r$, $R = 2/r^2$ everywhere.

The Weyl tensor

The Riemann tensor splits into two pieces: - The Ricci part (determined by matter via Einstein’s equations) - The Weyl tensor $C^\rho{}_{\sigma\mu\nu}$ (the trace-free part — 10 components)

The Weyl tensor describes curvature that exists even in vacuum — tidal forces, gravitational waves. In empty space ($T_{\mu\nu} = 0$), the Ricci tensor vanishes but the Weyl tensor can be nonzero. This is why gravity has effects far from any matter.

Flat spacetime means zero Riemann

If and only if $R^\rho{}{\sigma\mu\nu} = 0$ everywhere, the spacetime is flat — you can find coordinates in which $g{\mu\nu} = \eta_{\mu\nu}$ globally. The Riemann tensor is the definitive test for curvature.