The (1,3)-tensor R(X,Y)Z = ∇_X∇_Y Z − ∇_Y∇_X Z − ∇_{[X,Y]} Z measuring failure of commutativity of covariant derivatives. Contracts to Ricci and scalar curvature; vanishes iff M is locally flat.
Riemann curvature tensor
Related concepts
- Riemannian metric
- Gauss–Bonnet theorem
- FLRW flat matter-dominated solution (cosmology)
- Riemann R^ρ_{σμν}=∂_μΓ^ρ_{νσ}−∂_νΓ^ρ_{μσ}+ΓΓ−ΓΓ; (4,0) symmetries
- Bianchi I (algebraic): R^ρ_{[σμν]}=0; II (differential): ∇_{[λ}R_{|σ|μν]}=0
- S² round: R^θ_{φθφ} = sin²θ; Ricci scalar R = 2; antisym residual 0
- First Bianchi R^ρ_{[σμν]}=0 on S²: cyclic-sum scan ≡ 0