In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σ_p) depends on a two-dimensional plane σ_p in the tangent space at p. It is the Gaussian curvature of that section — the surface which has the plane σ_p as a tangent plane at p, obtained from geodesics which start at p in the directions of σ_p (in other words, the image of σ_p under the exponential map at p). Formally, the sectional curvature is a smooth real-valued function on the 2-Grassmannian bundle over the manifold.

The sectional curvature determines the curvature tensor completely, and is very useful 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
• positive curvature +1, elliptic geometry

The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit 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 .

Properties

• A complete Riemannian manifold has non-negative sectional curvature if and only if the function f_p(x) = dist²(p,x) is 1-concave for all points p.
• A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function f_p(x) = dist²(p,x) is 1-convex.

See also

• Riemann curvature tensor
• curvature of Riemannian manifolds
• curvature