Curvature — Riemann, Ricci, and Sectional

Curvature measures the failure of a manifold to be flat. The Riemann curvature tensor

R^{l}_{ij k}

encodes all curvature information; its contraction gives the Ricci tensor

Ric_{ij}

(average over directions), and the sectional curvature

K (σ)

records the Gaussian curvature of 2-dimensional slices. Applies to Riemannian and pseudo-Riemannian geometry, general relativity, and global analysis. Foundational work by Riemann (1854), Ricci-Curbastro & Levi-Civita (1900), and Cartan (1920s).

On a smooth manifold with a metric $g$ and the Levi-Civita connection $\nabla$ , curvature is the obstruction to the commutativity of second covariant derivatives. The full obstruction is the Riemann tensor, a $(1, 3)$ -tensor that, when lowered, satisfies rich algebraic symmetries reducing its independent components to $n^{2} (n^{2} - 1) /12$ . Tracing the Riemann tensor produces the Ricci tensor and, further, the scalar curvature; restricting to 2-planes produces sectional curvature, from which the entire Riemann tensor can be reconstructed. The surprising fact: in dimension 2 all three notions coincide (the Gaussian curvature), in dimension 3 the Ricci tensor determines Riemann completely, and only from dimension 4 onward does the Weyl tensor — the "conformal," trace-free remainder — carry genuinely independent information.

The Core Insight

Curvature is the non-commutativity of parallel transport. If you carry a vector around an infinitesimal closed loop spanned by directions $X$ and $Y$ , it returns rotated by an amount linear in the loop area; the rotation operator is precisely $R (X, Y)$ . Everything else — Ricci as an average rotation, sectional curvature as the rotation per unit area in a 2-plane — is a contraction or restriction of this single object.

Formal Statement

Riemann Curvature Tensor

Let $(M, g)$ be a smooth Riemannian manifold with Levi-Civita connection $\nabla$ . The Riemann curvature endomorphism is the $(1, 3)$ -tensor field

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

⟳postulate the simplest tensorial measure of the failure of

\nabla_{X} \nabla_{Y}

to commute; the

\nabla_{[X, Y]}

term is forced so the expression is

C^{\infty} (M)

-linear in all three arguments.

In coordinates with $\partial_{i} = \partial / \partial x^{i}$ and Christoffel symbols $Γ_{ij}^{k}$ ,

R^{l}_{ij k} = \partial_{j} Γ_{ik}^{l} - \partial_{k} Γ_{ij}^{l} + Γ_{ik}^{m} Γ_{j m}^{l} - Γ_{ij}^{m} Γ_{k m}^{l},

⟳expand

R (\partial_{j}, \partial_{k}) \partial_{i}

using

\nabla_{\partial_{j}} \partial_{i} = Γ_{j i}^{l} \partial_{l}

and the vanishing of

[\partial_{j}, \partial_{k}]

The fully covariant tensor is $R_{l ij k} = g_{l m} R^{m}_{ij k}$ , where symbols are: $g_{ij}$ the metric components, $Γ_{ij}^{k} = \frac{1}{2} g^{k m} (\partial_{i} g_{j m} + \partial_{j} g_{im} - \partial_{m} g_{ij})$ the connection coefficients, and indices run $1, \dots, n = dim M$ .

Sectional Curvature

For a 2-dimensional subspace (tangent 2-plane) $σ \subset T_{p} M$ spanned by linearly independent vectors $u, v$ , the sectional curvature is

K (σ) = \frac{⟨ R ( u , v ) v , u ⟩}{⟨ u , u ⟩ ⟨ v , v ⟩ - ⟨ u , v ⟩ ^{2}},

⟳normalize by the squared area of the parallelogram spanned by

u, v

(Gram determinant) so the result is independent of the chosen basis of

σ

It equals the Gaussian curvature at $p$ of the surface swept out by geodesics tangent to $σ$ .

Ricci and Scalar Curvature

The Ricci tensor is the trace of the Riemann endomorphism over its first and last slots:

Ric_{ij} = R^{k}_{ik j} = g^{k l} R_{k i l j},

⟳contract the

(1, 3)

tensor on the upper index and the third lower index; symmetry

Ric_{ij} = Ric_{j i}

follows from the pair-symmetry of

R

The scalar curvature is its metric trace:

S = g^{ij} Ric_{ij} = g^{ij} g^{k l} R_{k i l j} .

⟳contract Ricci once more with the inverse metric; in 2D this gives

S = 2 K

Geometrically, $Ric (u, u)$ is (up to a factor) the average sectional curvature over all 2-planes containing $u$ .

Algebraic Symmetries of Riemann

The covariant Riemann tensor satisfies, at every point:

R_{l ij k} = - R_{i l j k} = - R_{l ik j} = R_{j k l i}, R_{l ij k} + R_{l j k i} + R_{l k ij} = 0.

⟳antisymmetry in the last pair is built into

[1]

; antisymmetry in the first pair uses metric compatibility

\nabla g = 0

; the cyclic (first Bianchi) identity uses the torsion-free property; pair-symmetry follows by combining all three.

These reduce the number of independent components to $n^{2} (n^{2} - 1) /12$ (e.g. $1$ in 2D, $6$ in 3D, $20$ in 4D). The contracted second Bianchi identity yields

\nabla^{i} (Ric_{ij} - \frac{1}{2} S g_{ij}) = 0,

⟳contract

\nabla_{[m} R_{l ij] k} = 0

twice with

g

; the divergence-free combination is the Einstein tensor

G_{ij}

Why It Holds / Derivation

Tensoriality and the symmetries

Step 1 (Tensoriality). Replace

X \mapsto f X

[1]

for

f \in C^{\infty} (M)

. The Leibniz expansions of

\nabla_{f X} \nabla_{Y} Z

and

\nabla_{[f X, Y]} Z

produce derivative-of-

f

terms; one checks they cancel exactly, so

R (X, Y) Z

depends only on the pointwise values of

X, Y, Z

. Hence

R

is a genuine tensor. Step 2 (Antisymmetry in $X, Y$ ). Immediate from

[1]

since swapping

X \leftrightarrow Y

flips the commutator. Step 3 (Skew in last pair after lowering). Using metric compatibility

X ⟨ Y, Z ⟩ = ⟨ \nabla_{X} Y, Z ⟩ + ⟨ Y, \nabla_{X} Z ⟩

, differentiate twice and antisymmetrize to obtain

⟨ R (X, Y) Z, W ⟩ = - ⟨ R (X, Y) W, Z ⟩

. Step 4 (First Bianchi). Sum

[1]

cyclically over

X, Y, Z

. The torsion-free condition

\nabla_{X} Y - \nabla_{Y} X = [X, Y]

collapses the sum to the Jacobi identity for the bracket, which vanishes. Step 5 (Pair symmetry). Algebraically combine the three identities of Steps 2–4 (a standard four-term manipulation) to deduce

R_{l ij k} = R_{j k l i}

. Step 6 (Second Bianchi). In normal coordinates at

p

where

Γ = 0

(but

\partial Γ \neq = 0

[2]

reduces to

R^{l}_{ij k} = \partial_{j} Γ_{ik}^{l} - \partial_{k} Γ_{ij}^{l}

. Cyclic differentiation gives

\nabla_{m} R_{l ij k} + \nabla_{j} R_{l imk} + \nabla_{k} R_{l j mi} = 0

. Contracting twice with

g

produces

[7]

■

Bernhard Riemann | 1826–1866

In his 1854 habilitation lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen, Riemann introduced the notion of a manifold equipped with a quadratic line element $d s^{2} = g_{ij} d x^{i} d x^{j}$ and defined the sectional curvature via a six-component object — the seed of the Riemann tensor. He showed that a space is locally Euclidean if and only if this curvature vanishes identically, decades before tensor calculus existed to express it cleanly. His vision that geometry is determined by an intrinsic, possibly variable, metric — and his speculation that physical space might be curved — directly anticipated Einstein's general relativity sixty years later.

Building Intuition

Think of three nested pictures.

(a) Holonomy. Parallel-transport a vector around a tiny rectangle with sides

ϵ X

and

ϵ Y

. It comes back rotated by

- ϵ^{2} R (X, Y) + O (ϵ^{3})

. Riemann curvature is this rotation per unit area, as an operator on the tangent space. (b) Geodesic deviation. Two nearby geodesics with separation field

J

(a Jacobi field) accelerate toward or away from each other according to

\ddot{J} = - R (J, \overset{γ}{˙}) \overset{γ}{˙}

. Positive sectional curvature focuses geodesics (sphere); negative curvature spreads them apart (hyperbolic plane). (c) Volume of small balls. Ricci curvature controls how the volume of a geodesic ball deviates from the Euclidean value:

vol (B_{r} (p)) = ω_{n} r^{n} (1 - \frac{S}{6 ( n + 2 )} r^{2} + \dots)

, and the directional refinement involves

Ric

. Positive Ricci means balls are smaller than Euclidean — matter, in the GR analogy, concentrates volume.

Geodesic deviation: parallel geodesics converge on a positively curved surface

Worked Examples

The round 2-sphere of radius a

Given:

S_{a}^{2}

with metric

d s^{2} = a^{2} (d θ^{2} + sin^{2} θ d φ^{2})

. Compute Riemann, Ricci, scalar, and sectional curvature. Step 1 — Christoffels. With

g_{θ θ} = a^{2}

g_{φφ} = a^{2} sin^{2} θ

: the nonzero symbols are

Γ_{φφ}^{θ} = - sin θ cos θ

Γ_{θ φ}^{φ} = Γ_{φθ}^{φ} = cot θ

. Step 2 — Riemann. Using

[2]

R^{θ}_{φθ φ} = \partial_{θ} Γ_{φφ}^{θ} - Γ_{φθ}^{φ} Γ_{φφ}^{θ} = (cos^{2} θ - sin^{2} θ) + cot θ sin θ cos θ = sin^{2} θ .

Lowering:

R_{θ φθ φ} = g_{θ θ} R^{θ}_{φθ φ} = a^{2} sin^{2} θ .

Step 3 — Sectional curvature. The only 2-plane is

T_{p} S^{2}

. Gram determinant

= g_{θ θ} g_{φφ} = a^{4} sin^{2} θ

. Then by

[3]

K = \frac{R _{θ φθ φ}}{a ^{4} sin ^{2} θ} = \frac{a ^{2} sin ^{2} θ}{a ^{4} sin ^{2} θ} = \frac{1}{a ^{2}} .

Step 4 — Ricci and scalar.

Ric_{ij} = K g_{ij} = \frac{1}{a ^{2}} g_{ij}

(Einstein manifold);

S = g^{ij} Ric_{ij} = \frac{2}{a ^{2}}

K = \frac{1}{a ^{2}} (units m^{- 2}), S = \frac{2}{a ^{2}}, Ric = \frac{1}{a ^{2}} g .

For

a = 2 m

K = 0.25 m^{- 2}

S = 0.5 m^{- 2}

Hyperbolic plane in upper-half-plane model

Given:

H^{2}

with

d s^{2} = \frac{d x ^{2} + d y ^{2}}{y ^{2}}

y > 0

. Find

K

. Step 1 — Metric & Christoffels.

g_{xx} = g_{y y} = y^{- 2}

. Nonzero:

Γ_{x y}^{x} = Γ_{y x}^{x} = - 1/ y

Γ_{xx}^{y} = 1/ y

Γ_{y y}^{y} = - 1/ y

. Step 2 — Riemann. Computing

R^{x}_{y x y}

via

[2]

R^{x}_{y x y} = \partial_{x} Γ_{y y}^{x} - \partial_{y} Γ_{y x}^{x} + Γ_{x m}^{x} Γ... = - 1/ y^{2} .

Lowering:

R_{x y x y} = g_{xx} R^{x}_{y x y} = y^{- 2} \cdot (- 1/ y^{2}) \cdot (- 1)

→ carry signs carefully →

R_{x y x y} = - 1/ y^{4}

. Step 3 — Sectional curvature. Gram determinant

= g_{xx} g_{y y} = y^{- 4}

. Then

K = \frac{R _{x y x y}}{y ^{- 4}} = \frac{- y ^{- 4}}{y ^{- 4}} = - 1.

K \equiv - 1 (constant negative, units length^{- 2}) .

Thus

H^{2}

is the model space of constant negative curvature; geodesics diverge exponentially with separation rate

\sim e^{s}

Common Misconceptions

"Ricci curvature is just the average of the Riemann tensor and so loses no information." False in general. In dimensions

\geq 4

the Weyl tensor — the trace-free part of Riemann — is invisible to Ricci. A Ricci-flat manifold (

Ric = 0

) can be highly curved: Schwarzschild spacetime and Calabi–Yau manifolds are Ricci-flat yet not flat. Only in dimensions 2 and 3 does Ricci (or scalar in 2D) determine the full Riemann tensor. "Positive scalar curvature means positive sectional curvature everywhere." No. Scalar curvature is a double average; it can be positive while some sectional curvatures are negative. The product

S^{2} \times H^{2}

(suitably scaled) can have positive scalar curvature with negative sectional curvatures present.

Flat torus vs. round sphere — same topology of "closed surface," opposite curvature

The flat torus $T^{2} = R^{2} / Z^{2}$ has $K \equiv 0$ everywhere yet is a compact closed surface. The round sphere $S^{2}$ has $K = 1/ a^{2} > 0$ . Both are closed orientable surfaces, but by Gauss–Bonnet $\int_{M} K d A = 2 π χ (M)$ : the torus has $χ = 0$ forcing total curvature zero, while $S^{2}$ has $χ = 2$ forcing $\int K d A = 4 π$ . This shows curvature is not a topological invariant pointwise, but its integral is constrained by topology — a recurring theme.

Connections

The Riemann tensor sits at the intersection of several structures. Its definition rests on the Levi-Civita Connection and the notion of Parallel Transport and Holonomy; its symmetries encode the Bianchi Identities. Contraction yields Ricci, whose vanishing-with-cosmological-constant is the content of the Einstein Field Equations. The trace-free remainder is the Weyl Conformal Tensor, governing conformal geometry. Globally, sectional curvature sign controls comparison theorems linking to Jacobi Fields and Geodesic Deviation and to Gauss–Bonnet Theorem in 2D. Curvature flows such as Ricci Flow evolve the metric by $\partial_{t} g = - 2 Ric$ , the engine of Perelman's proof of the Poincaré Conjecture.

Real-World Applications

General Relativity. The Einstein equation $G_{ij} = Ric_{ij} - \frac{1}{2} S g_{ij} = 8 π T_{ij}$ equates spacetime Ricci/scalar curvature to the stress-energy tensor; gravitational lensing and GPS time corrections are curvature effects measured daily.
Cosmology. The sign of spatial sectional curvature ( $k = + 1, 0, - 1$ ) in FLRW models distinguishes a closed, flat, or open universe; CMB measurements constrain it near zero.
Geometric data analysis & manifold learning. Estimating Ricci curvature of data graphs (Ollivier–Ricci curvature) detects community structure and bottlenecks in networks.
Computer graphics & mesh processing. Discrete Gaussian/mean curvature drives surface smoothing, remeshing, and the "curvature flow" denoising of 3D scans.
Materials science. Curvature of crystalline membranes and graphene sheets governs buckling and electronic transport; Gaussian curvature relates to disclination defects.
Robotics & optimal control. Curvature of configuration-space Riemannian metrics determines stability of geodesic motion planning and the conjugate-point structure of trajectories.
Medical imaging. Cortical surface curvature quantifies brain folding (gyrification) for diagnosing developmental disorders.

Historical Development

1827 — Gauss's Disquisitiones generales circa superficies curvas proves the Theorema Egregium: Gaussian curvature is intrinsic, depending only on the metric.
1854 — Riemann's habilitation lecture introduces $n$ -dimensional manifolds, the metric line element, and a sectional-curvature object generalizing Gauss.
1869 — Christoffel introduces the symbols $Γ_{ij}^{k}$ and the covariant derivative machinery enabling explicit curvature computation.
1888–1896 — Bianchi derives the differential identities (the second Bianchi identity) governing the Riemann tensor.
1900 — Ricci-Curbastro and Levi-Civita publish Méthodes de calcul différentiel absolu, founding tensor calculus and naming the Ricci contraction.
1915 — Einstein, with Hilbert, formulates general relativity, placing the Ricci/Einstein tensor at the heart of gravitation.
1917 — Levi-Civita defines parallel transport, revealing curvature as holonomy.
1920s — Élie Cartan develops the moving-frame/exterior-calculus formulation $Ω = d ω + ω \land ω$ and the theory of symmetric spaces.
1950s–80s — Comparison geometry matures (Rauch, Berger, Cheeger–Gromoll, Toponogov); the sphere theorem and splitting theorems link curvature bounds to topology.
1982–2003 — Hamilton's Ricci flow and Perelman's resolution of the Poincaré and geometrization conjectures.

Subtleties & Limits

Pinched curvature and topology. The differentiable sphere theorem (Brendle–Schoen, 2009) shows that a simply connected manifold with pointwise

\frac{1}{4}

-pinched sectional curvature is diffeomorphic to a sphere — but the optimal constants and the analogous statements for Ricci pinching remain partly open. Positive scalar curvature. Classifying which manifolds admit metrics of positive scalar curvature is unresolved in general; the Gromov–Lawson and Stolz results handle large families via index theory, yet a complete topological characterization is missing. Ricci-flat with special holonomy. The existence and moduli of compact

G_{2}

- and

Spin (7)

-holonomy Ricci-flat manifolds, and the explicit form of generic Calabi–Yau metrics, are still beyond closed-form reach. Discrete curvature. No single discretization of Ricci curvature simultaneously satisfies all desirable convergence and Bianchi-type identities; reconciling Ollivier, Forman, and Regge-calculus notions is an active frontier.