# Sectional Curvature

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

### Other articles related to "sectional curvature, curvature":

Manifolds With Positive Sectional Curvature
... the Hopf conjecture on whether there is a metric of positive sectional curvature on ) ... most typical way of constructing new examples is the following corollary from the O'Neill curvature formulas if is a Riemannian manifold admitting a free isometric ...
Classical Theorems in Riemannian Geometry - Geometry in Large - Sectional Curvature Bounded Above
... connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R^n with n = dim M via the exponential map at any point ... that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic ... If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT(k) space ...
Minimal Volume - Definition
... Let represent the sectional curvature ... of the volume of over all metrics with bounded sectional curvature ... The inclusion of bounds on sectional curvature suffices, as ...
Scalar Curvature - Special Cases - Space Forms
... form is by definition a Riemannian manifold with constant sectional curvature ... space vanishes identically, so the scalar curvature does as well ... n-spheres The sectional curvature of an n-sphere of radius r is K = 1/r2 ...

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