Riemann Curvature Tensor — Mathematics Knowledge System

Overview

The Riemann curvature tensor is the fundamental measure of curvature of a Riemannian manifold. It captures the failure of parallel transport to be path-independent: if you transport a vector around a small parallelogram, it returns rotated by an amount proportional to the curvature. The Riemann tensor encodes all local geometric information about the manifold beyond what is given by the metric, and is central to both Riemannian geometry and Einstein's general theory of relativity (where the Einstein tensor, derived from the Riemann tensor, is set equal to the stress-energy tensor).

Intuition

Imagine parallel-transporting a vector around a small closed loop on a surface. On flat ℝ², the vector returns unchanged. On a curved surface, it comes back rotated. The Riemann curvature tensor measures this rotation infinitesimally. More precisely, R(X,Y)Z measures the difference between ∇_X∇_Y Z and ∇_Y∇_X Z — the extent to which covariant derivatives fail to commute. Non-commutativity is precisely curvature.

Formal Definition

Definition

The Riemann curvature tensor R is a (1,3) tensor defined by the commutator of covariant derivatives. Its components R^l_{ijk} in local coordinates are given by derivatives and products of Christoffel symbols.

R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z

Riemann curvature tensor (coordinate-free)

R^l{}_{kij} = \partial_i \Gamma^l_{jk} - \partial_j \Gamma^l_{ik} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik}

Components in local coordinates

R_{ijkl} = g_{im} R^m{}_{jkl}

Fully covariant Riemann tensor (all indices down)

K(\sigma) = \frac{R(X,Y,Y,X)}{g(X,X)g(Y,Y) - g(X,Y)^2}

Sectional curvature of a 2-plane σ = span{X, Y}

R_{ij} = R^k{}_{ikj} = \sum_k R^k{}_{ikj}

Ricci tensor (contraction of Riemann tensor)

R = g^{ij} R_{ij}

Scalar curvature

Properties

Flat manifold characterisation

R \equiv 0 \iff \text{M is locally isometric to Euclidean space}

Space forms

\text{Constant sectional curvature } K: \quad K > 0 \sim S^n, \quad K = 0 \sim \mathbb{R}^n, \quad K < 0 \sim H^n

Einstein tensor

G_{ij} = R_{ij} - \frac{1}{2} R g_{ij}, \quad \nabla^i G_{ij} = 0

Theorems

Theorem 1: Symmetries of the Riemann Tensor

R_{ijkl} = -R_{jikl} = -R_{ijlk} = R_{klij}, \quad R_{ijkl} + R_{iklj} + R_{iljk} = 0

Theorem 2: Second Bianchi Identity

\nabla_m R_{ijkl} + \nabla_k R_{ijlm} + \nabla_l R_{ijmk} = 0

Theorem 3: Schur's Lemma

\text{If } M \text{ is a connected Riemannian manifold of dimension} \geq 3 \text{ with constant sectional curvature, then } M \text{ is a space form (has constant } K \text{ everywhere).}

Worked Examples

1
The round sphere S^n of radius r in ℝ^{n+1} has constant sectional curvature K = 1/r².
$K = \frac{1}{r^2}$
2
To verify for S² (r = 1): the Riemann tensor components can be computed from the Christoffel symbols. For any orthonormal frame {e₁, e₂}:
$R(e_1, e_2, e_2, e_1) = g(R(e_1, e_2)e_2, e_1)$
3
Using the formula for S² with the round metric g = dθ² + sin²θ dφ², the curvature component R^θ_{φθφ} = sin²θ.
$R^\theta{}_{\phi\theta\phi} = \sin^2\theta$
4
Sectional curvature K = R_{θφθφ}/(g_{θθ}g_{φφ} − g_{θφ}²) = sin²θ/(1·sin²θ − 0) = 1.
$K = 1$

✓ Answer

S^n of radius r has constant sectional curvature K = 1/r². For the unit sphere, K = 1.

Practice Problems

Hardproof writing

Prove that a manifold with R ≡ 0 (flat manifold) has locally commuting parallel transport: parallel transport around any sufficiently small loop is the identity.

Hardfree response

How many independent components does the Riemann tensor have on an n-dimensional manifold?

Common Mistakes

Common Mistake

Confusing the Riemann tensor, Ricci tensor, and scalar curvature

They are different: Rᵃᵦᵧᵟ is (1,3), Rᵃᵦ = Rᵧᵃᵧᵦ is a contraction (0,2), and R = gᵃᵦRᵃᵦ is a scalar. They carry progressively less information.

Common Mistake

Zero Ricci curvature means flat

Ricci-flat manifolds (Rᵢⱼ = 0) need not be flat — they can have non-zero Riemann tensor. Calabi–Yau manifolds are famous examples: Ricci-flat but highly curved.

Quiz

Historical Background

Riemann hinted at the curvature concept in his 1854 lecture through the sectional curvature formula. Christoffel (1869) and Riemann himself (posthumous, 1868) gave explicit curvature expressions. Ricci developed the contracted tensors (Ricci tensor, scalar curvature) as part of his tensor calculus. Einstein and Grossmann (1913–1915) identified the Riemann curvature tensor as the proper object encoding gravity. The Bianchi identity, discovered by Luigi Bianchi in 1902, plays a role analogous to Maxwell's equations' Jacobi identity and is crucial for deriving the Einstein equations.

1854/1868
Riemann introduces sectional curvature; posthumous publication of explicit formula
Bernhard Riemann
1869
Christoffel derives the curvature tensor explicitly in components
Elwin Bruno Christoffel
1902
Bianchi discovers the differential identity ∇R + cyclic = 0
Luigi Bianchi
1915
Einstein uses the Ricci tensor and scalar curvature in the field equations of general relativity
Albert Einstein

Summary

The Riemann tensor R(X,Y)Z = ∇_X∇_Y Z − ∇_Y∇_X Z − ∇_{[X,Y]} Z measures the curvature of a Riemannian manifold.
Its components Rˡₖᵢⱼ involve second derivatives of the metric (through Christoffel symbols).
Key contractions: Ricci tensor Rᵢⱼ and scalar curvature R; these appear in Einstein's field equations.
The Bianchi identities (first and second) are differential constraints on the curvature, analogous to Maxwell's equations.
Constant sectional curvature: K > 0 (spheres), K = 0 (Euclidean), K < 0 (hyperbolic space).

References

BookMisner, C., Thorne, K., Wheeler, J. — Gravitation, W.H. Freeman, 1973, Chapter 11
Bookdo Carmo, M. P. — Riemannian Geometry, Birkhäuser, 1992, Chapter 4
WebsiteWikipedia — Riemann curvature tensor
WebsiteWolfram MathWorld — Riemann Tensor