Sectional curvature

In differential geometry, the sectional curvature K(σ) depends on a two-dimensional plane σ. It is half of the scalar curvature of that section - the 2-dimensional submanifold which has the plane σ as a tangent plane at a given point, obtained from geodesics which lie in σ - the image of a plane under the exponential map of a neighbourhood of a point.

It determines the curvature tensor completely, and is a way to understand it in terms of more intuitive curvature of surface sections - in terms of intrinsic Gaussian curvature.

Riemannian manifolds with constant sectional curvature are the most simple. By rescaling the metric there are three possible cases - negative curvature -1 - hyperbolic geometry, zero curvature - Euclidean geometry, or positive curvature +1 - elliptic geometry. The model manifolds for the three geometries are hyperbolic space, Euclidean space and a sphere. They are the only complete, simply connected Riemannian manifolds of given sectional curvature, and all other complete constant curvature manifolds are quotients of those by some group of isometries .