Sectional curvature

Sectional curvature measures the Gaussian curvature of a two-dimensional surface obtained by slicing a Riemannian manifold through a point with a plane spanned by two tangent vectors. It is the most detailed of the classical curvature invariants — Ricci curvature and scalar curvature are both averages of sectional curvature over different subspaces — and it determines the full Riemann curvature tensor. A manifold has constant sectional curvature if and only if it is a space form: Euclidean space, a sphere, or hyperbolic space. The Cartan-Hadamard theorem states that a complete, simply connected manifold with non-positive sectional curvature is diffeomorphic to Euclidean space, one of the most powerful topological constraints in geometry. The surprise is that sectional curvature, despite being the most local and detailed invariant, often fails to control global structure where Ricci curvature succeeds: the difference between local detail and global control is one of the central tensions in modern geometry.