Riemann curvature tensor

The Riemann curvature tensor is the fundamental object of Riemannian geometry that measures the failure of vector parallel transport to be path-independent. Unlike the Ricci and scalar curvatures, which are contractions that discard information, the full Riemann tensor captures tidal forces and gravitational radiation — the degrees of freedom that vanish in vacuum solutions of the Einstein field equations yet govern how nearby geodesics converge or diverge. Its components R^ρ_σμν encode whether a manifold can be flattened locally, and the tensor's algebraic symmetries reduce its n⁴ apparent components to n²(n²−1)/12 independent degrees of freedom.

The Riemann tensor is not merely a geometric invariant. It is the obstruction to integrability: a manifold is locally isometric to Euclidean space if and only if its Riemann tensor vanishes identically. This makes it the curvature in its most complete form, and every other curvature — Ricci, scalar, sectional — is a partial view through a narrower aperture. The persistent tendency in physics to work with the Einstein tensor rather than the full Riemann tensor is not a simplification but a conceptual narrowing: it treats gravity as sourced by matter while treating the propagating degrees of freedom as secondary. The Riemann tensor says they are primary.