The Sectional curvature reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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 surface 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 it 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, 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 .

Properties

1. A comlete Riemannian manifold has non-negative sectional curvature if and only if the function is 1-concave for all points p.
2. A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function is 1-convex.

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