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The sectional curvature K(σ) of a 2-plane σ ⊂ TpM is the Gaussian curvature of the surface formed by geodesics tangent to σ. It is the finest pointwise curvature invariant and determines the full Riemann tensor. Spaces of constant sectional curvature — Euclidean space (K=0), spheres (K>0), and hyperbolic space (K<0) — are the model geometries of Riemannian geometry. Comparison theorems use sectional curvature bounds to control geodesic behavior, volume growth, and topology.
The Riemann curvature tensor is a complicated object — it takes four vector inputs. Sectional curvature distills this to a function on 2-planes, which is both more geometric and more tractable. Given a 2-plane σ ⊂ TpM, consider the surface Σ traced out by geodesics starting at p with initial velocities in σ. The sectional curvature K(σ) is the Gaussian curvature of this surface at p. It measures how fast geodesics in the plane σ converge or diverge: K > 0 means geodesics converge (like on a sphere), K < 0 means they diverge (like in hyperbolic space), and K = 0 means they stay parallel (like in flat space).
The three space forms — Euclidean space ℝⁿ, the sphere Sⁿ, and hyperbolic space Hⁿ — are the complete, simply connected Riemannian manifolds of constant sectional curvature K = 0, K > 0, and K < 0 respectively. These are the "maximally symmetric" geometries: each admits an isometry group of dimension n(n+1)/2, the maximum possible. The classification theorem says these are the only possibilities for constant curvature. Every complete manifold of constant curvature is a quotient of one of these by a discrete group of isometries — the flat torus is ℝ²/ℤ², the real projective space is Sⁿ/ℤ₂, and hyperbolic surfaces are H²/Γ for various Fuchsian groups Γ.
Comparison geometry uses sectional curvature bounds to control the behavior of geodesics and volumes. If K ≤ κ (curvature bounded above), then geodesic triangles are "thinner" than in the model space of constant curvature κ — geodesics spread apart at least as fast. If K ≥ κ (curvature bounded below), geodesic triangles are "fatter." The Rauch comparison theorem makes this precise for Jacobi fields, and the Toponogov triangle comparison theorem extends it to global distance comparisons. These tools yield deep topological results: the sphere theorem (manifolds with ¼ < K ≤ 1 are homeomorphic to spheres), the Cartan-Hadamard theorem (complete manifolds with K ≤ 0 have contractible universal cover), and the soul theorem for non-negative curvature.
The relationship between sectional, Ricci, and scalar curvature forms a hierarchy of information. Constant sectional curvature is the strongest condition (the space form classification). Positive sectional curvature implies positive Ricci, which implies positive scalar curvature — but the converses are false. Each level of the hierarchy gives progressively weaker geometric and topological constraints. Understanding this hierarchy and finding optimal conditions for geometric conclusions is one of the driving programs in Riemannian geometry.