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 Incoming links: Riemann curvature tensor

Ricci tensor in two dimensions Created 2026-09-24 Updated 2026-10-07

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and force

R_{α β} = \frac{1}{2} g_{α β} R .

Consequently the Einstein tensor vanishes identically.

n \geq 3

S_{ij} = R_{ij} - R g_{ij} /4

, and the full Riemann curvature tensor is reconstructed from it.

R_{ij pq} = g_{i p} S_{j q} - g_{i q} S_{j p} - g_{j p} S_{i q} + g_{j q} S_{i p}

, where

S

is the Schouten tensor. Consequently every three-dimensional Ricci-flat Riemannian manifold is a flat Riemannian manifold. The implication fails in higher dimensions.

D > 2

R = 2 D Λ/ (D - 2)

and hence

R_{ab} = 2Λ g_{ab} / (D - 2)

; in four dimensions this reduces to

R_{ab} = Λ g_{ab}

Weyl tensor Created 2026-09-24 Updated 2026-09-28

The Weyl tensor is the completely trace-free part of the Riemann curvature tensor. In four dimensions,

C_{α β γ δ} = R_{α β γ δ} - \frac{1}{2} (g_{α γ} R_{β δ} + g_{β δ} R_{α γ} - g_{α δ} R_{β γ} - g_{β γ} R_{α δ}) + \frac{R}{6} (g_{α γ} g_{β δ} - g_{α δ} g_{β γ}) .