Ricci and Scalar Curvature — Mathematics Knowledge System

Overview

Ricci curvature is a contraction of the Riemann curvature tensor that captures how volumes in a Riemannian manifold deviate from Euclidean volumes. Scalar curvature is the further contraction to a single real-valued function on the manifold. These invariants play a central role in Riemannian geometry — Ricci curvature governs geodesic focusing via the Raychaudhuri equation, drives the Ricci flow, and appears prominently in Einstein's field equations of general relativity.

Intuition

Imagine shooting a spray of geodesics from a point in all directions. In flat space, the cross-sectional area of the spray grows like r^(n-1). Positive Ricci curvature causes geodesics to converge faster — the spray narrows — while negative Ricci curvature causes divergence. Scalar curvature gives a single number summarising all these effects simultaneously.

Formal Definition

Definition

On an n-dimensional Riemannian manifold (M, g), the Riemann curvature tensor R is a (1,3)-tensor. The Ricci tensor Ric is obtained by tracing over the first and third indices. Scalar curvature R (the Ricci scalar) is the metric trace of the Ricci tensor.

\mathrm{Ric}_{ij} = R^k{}_{ikj} = \sum_{k=1}^n R^k{}_{ikj}

Ricci tensor — contraction of Riemann tensor

\mathrm{Ric}_{ij} = \partial_k \Gamma^k_{ij} - \partial_j \Gamma^k_{ik} + \Gamma^k_{k\ell}\Gamma^\ell_{ij} - \Gamma^k_{j\ell}\Gamma^\ell_{ik}

Explicit Christoffel-symbol form

S = g^{ij}\,\mathrm{Ric}_{ij} = \sum_{i,j} g^{ij}\mathrm{Ric}_{ij}

Scalar curvature — metric trace of Ricci tensor

G_{ij} = \mathrm{Ric}_{ij} - \tfrac{1}{2}S\,g_{ij}

Einstein tensor — divergence-free combination

Notation

Notation	Meaning
$\mathrm{Ric},\; \mathrm{Ric}_{ij}$	Ricci curvature tensor
$S,\; R$	Scalar curvature (Ricci scalar)
$R^k{}_{ijl}$	Full Riemann curvature tensor
$G_{ij}$	Einstein tensor

Properties

Symmetry

\mathrm{Ric}_{ij} = \mathrm{Ric}_{ji}

Contracted Bianchi identity

\nabla^i G_{ij} = 0

Dimension 2: Ricci determines Riemann

\mathrm{Ric}_{ij} = K g_{ij} \text{ where } K \text{ is the Gaussian curvature}

Theorems

Theorem 1: Myers' Theorem

If (M, g) is a complete Riemannian manifold with \mathrm{Ric} \geq (n-1)k > 0, then M is compact with diameter \leq \pi / \sqrt{k} and finite fundamental group.

Theorem 2: Bishop–Gromov Volume Comparison

If \mathrm{Ric} \geq (n-1)k, the ratio \mathrm{Vol}(B(p,r)) / V_k(r) is non-increasing in r, where V_k(r) is the volume of a ball of radius r in the space form of curvature k.

Theorem 3: Schur's Lemma (curvature)

If \mathrm{Ric} = f\,g for some smooth function f on a connected Riemannian manifold of dimension n \geq 3, then f is constant and (M,g) is Einstein.

Worked Examples

1
The Riemann tensor of S^n(1) satisfies R_{ijkl} = g_{ik}g_{jl} - g_{il}g_{jk}.
$R_{ijkl} = g_{ik}g_{jl} - g_{il}g_{jk}$
2
Contracting to get Ricci: Ric_{ij} = (n-1)g_{ij}.
$\mathrm{Ric}_{ij} = (n-1)\,g_{ij}$
3
Taking the metric trace: S = g^{ij}(n-1)g_{ij} = (n-1)n.
$S = n(n-1)$

✓ Answer

S = n(n-1). For the 2-sphere: S = 2 (Gaussian curvature K = 1).

Practice Problems

Hardproof writing

Prove that any Einstein manifold (Ric = λg) of dimension ≥ 3 has constant scalar curvature.

Hardfree response

State and explain the significance of the Bishop–Gromov inequality in Riemannian geometry.

Common Mistakes

Common Mistake

Confusing Ricci curvature with sectional curvature

Sectional curvature K(σ) depends on a 2-plane σ ⊂ T_pM. Ricci curvature Ric(v,v) is the average of sectional curvatures over all 2-planes containing v.

Common Mistake

Thinking scalar curvature determines the full geometry

Scalar curvature is a single function; it contains far less information than the full Riemann tensor. Two manifolds can have the same scalar curvature but completely different geometries.

Quiz

The Ricci tensor Ric_{ij} is obtained from the Riemann tensor by:

Antisymmetrising over all indicesContracting the first and third indicesTaking the exterior derivativeSymmetrising over all pairs

Myers' theorem concludes compactness from:

Negative scalar curvatureUniformly positive Ricci curvatureZero sectional curvatureConstant scalar curvature

Historical Background

Gregorio Ricci-Curbastro developed the tensor calculus in the 1880s–1900s, providing the algebraic machinery from which the Ricci tensor is named. Tullio Levi-Civita collaborated to produce the 1900 paper 'Méthodes de calcul différentiel absolu et leurs applications'. Einstein adopted this framework in 1915 for general relativity. The modern study of Ricci curvature flourished with Myers' theorem (1941) and Hamilton's Ricci flow (1982), culminating in Perelman's proof of the Geometrization conjecture (2003).

1900
Ricci and Levi-Civita publish absolute differential calculus
Gregorio Ricci-Curbastro, Tullio Levi-Civita
1915
Einstein's field equations embed the Ricci tensor centrally
Albert Einstein
1941
Myers' theorem: positive Ricci curvature implies compact manifold
Sumner Myers
1982
Hamilton introduces Ricci flow
Richard Hamilton
2003
Perelman uses Ricci flow to prove Geometrization conjecture
Grigori Perelman

Summary

Ricci curvature Ric_{ij} is the trace of the Riemann tensor over its first and third indices.
Scalar curvature S = g^{ij} Ric_{ij} is the complete contraction to a function.
Positive Ricci curvature causes geodesic convergence and implies compactness (Myers).
Ricci flow ∂g/∂t = -2 Ric is Hamilton's heat-equation analogue for metrics, used by Perelman to prove the Poincaré conjecture.
The Einstein tensor G_{ij} = Ric_{ij} - (S/2)g_{ij} is divergence-free and appears in GR.

References

Bookdo Carmo, M.P. — Riemannian Geometry (1992), Chapters 4–5
BookPetersen, P. — Riemannian Geometry, 3rd ed. (2016), Chapter 7
BookChow, B. & Knopf, D. — The Ricci Flow: An Introduction (2004)
WebsiteWikipedia — Ricci curvature
WebsiteMathWorld — Ricci Tensor