Sectional Curvature — Mathematics Knowledge System

Overview

Sectional curvature is a real-valued function that assigns to each 2-dimensional tangent plane at a point of a Riemannian manifold a number measuring how curved the manifold is in that direction. It generalizes the Gaussian curvature of surfaces and is the most geometric component of the full Riemann curvature tensor. Constant sectional curvature spaces — spheres (K > 0), Euclidean space (K = 0), and hyperbolic space (K < 0) — are the model spaces of comparison geometry.

Intuition

At each point p of a Riemannian manifold, pick two orthonormal tangent vectors u and v. They span a 2-plane σ. Shoot out the geodesics from p in all directions in σ — these form a small surface. The sectional curvature K(σ) is the Gaussian curvature of that surface at p. So sectional curvature asks: 'how curved is the manifold in the σ direction?' Different 2-planes at p can give different sectional curvatures, and the full curvature tensor is determined by all of them.

Formal Definition

Definition

Let (M, g) be a Riemannian manifold with Riemann curvature tensor R. For a 2-dimensional subspace σ ⊆ T_pM spanned by linearly independent vectors u, v, the sectional curvature K(σ) is defined as follows.

K(\sigma) = K(u,v) = \frac{g(R(u,v)v,\, u)}{g(u,u)\,g(v,v) - g(u,v)^2}

The denominator is the squared area of the parallelogram spanned by u and v

Sectional curvature of plane σ = span{u,v}

K(u,v) = \frac{R_{ijkl}\, u^i v^j v^k u^l}{(g_{ij}g_{kl} - g_{il}g_{kj})\, u^i u^j v^k v^l}

In local coordinates

K(e_1, e_2) = g(R(e_1,e_2)e_2,\, e_1) \quad \text{for orthonormal } e_1, e_2

Simplified formula for orthonormal frame

Theorems

Theorem 1: Curvature Tensor from Sectional Curvatures

\text{The Riemann curvature tensor is completely determined by the sectional curvatures } K(u,v) \text{ for all pairs } u,v \in T_pM.

Theorem 2: Schur's Theorem

\text{If } K(u,v) \text{ is constant over all 2-planes at every point (isotropic) and } \dim M \geq 3, \text{ then } K \text{ is also constant over all points of } M.

Theorem 3: Space Forms

\text{A complete simply-connected Riemannian manifold of constant sectional curvature } K \text{ is isometric to: } S^n(1/\sqrt{K}) \text{ if } K>0, \; \mathbb{R}^n \text{ if } K=0, \; \mathbb{H}^n(1/\sqrt{|K|}) \text{ if } K<0.

Worked Examples

1
On S² with the round metric, use spherical coordinates (θ, φ). The Riemann tensor has the single independent component R_{θφθφ}. For the unit sphere, Gauss's theorem gives K = 1.
$K = 1$
2
Verify directly: take orthonormal vectors e₁ = ∂/∂θ, e₂ = (1/sin θ) ∂/∂φ at a point. Compute R(e₁, e₂)e₂ using the Christoffel symbols Γ^θ_{φφ} = −sin θ cos θ and Γ^φ_{θφ} = cot θ.
$\Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta, \quad \Gamma^\phi_{\theta\phi} = \cot\theta$
3
A direct computation (using the formula R^i_{jkl} = ∂_k Γ^i_{lj} − ∂_l Γ^i_{kj} + Γ^i_{km}Γ^m_{lj} − Γ^i_{lm}Γ^m_{kj}) gives R_{θφθφ} = sin²θ.
$R_{\theta\phi\theta\phi} = \sin^2\theta$
4
The sectional curvature formula: K = R_{θφθφ} / (g_{θθ}g_{φφ} − g_{θφ}²). With g_{θθ} = 1, g_{φφ} = sin²θ, g_{θφ} = 0:
$K = \frac{\sin^2\theta}{1 \cdot \sin^2\theta - 0} = 1$

✓ Answer

The unit sphere S² has constant sectional curvature K = 1 everywhere.

Practice Problems

Mediumfree response

For a product manifold M × N, express the sectional curvature of a 2-plane σ in terms of the sectional curvatures of M and N.

Mediumproof writing

Prove that K(u,v) is well-defined, i.e., it depends only on the 2-plane σ = span{u,v}, not on the specific basis chosen.

Quiz

Summary

Sectional curvature K(σ) assigns to each 2-plane σ ⊆ T_pM the number g(R(u,v)v,u)/|u∧v|², independent of basis choice.
It generalizes Gaussian curvature: for surfaces K(σ) = K_G is the Gaussian curvature.
The three model spaces of constant curvature are Sⁿ (K=1), ℝⁿ (K=0), ℍⁿ (K=−1).
Schur's theorem: isotropic sectional curvature in dim ≥ 3 implies globally constant K.
Sectional curvatures completely determine the full Riemann curvature tensor via the polarization identity.