Sectional curvature
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In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature <math>K(\sigma_p)<math> depends on a two-dimensional plane <math>\sigma_p<math> in the tangent space at p. It is the Gaussian curvature of that section — the surface which has the plane <math>\sigma_p<math> as a tangent plane at p, obtained from geodesics which start at p in the directions of <math>\sigma_p<math> (in other words, the image of <math>\sigma_p<math> 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 <math>f_p(x)=dist^2(p,x)<math> is 1-concave for all points p.
- A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function <math>f_p(x)=dist^2(p,x)<math> is 1-convex.