Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(sigma_p) depends on a two-dimensional plane sigma_p in the tangent space at p. It is the Gaussian curvature of that section — the surface which has the plane sigma_p as a tangent plane at p, obtained from geodesics which start at p in the directions of sigma_p (in other words, the image of sigma_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.

Related Topics:
Riemannian geometry - Curvature of Riemannian manifolds - Gaussian curvature - Surface - Geodesic - Exponential map - Grassmannian - Bundle

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The sectional curvature determines the curvature tensor completely, and is very useful geometric notion.

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Riemannian manifolds with constant sectional curvature are the most simple. By rescaling the metric there are three possible cases

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• 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 .

Related Topics:
Hyperbolic space - Euclidean space - Sphere - Complete - Simply connected - Isometries

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Sectional curvature: Properties ►