Sectional curvature

In Riemannian geometry, the sectional curvature depends on a two-dimensional plane σ in the tangent space at p. It is the Gauss curvature of that section - the 2-dimensional submanifold which has the plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of σ - the image of σ under the exponential map at p.

Sectional curvatures in all directions at p determine the curvature tensor completely, and in a way is more usefull geometric notion.

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 .