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. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. ... Curvature is the amount by which a geometric object deviates from being flat. ... An open surface with X-, Y-, and Z-contours shown. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

The sectional curvature determines the curvature tensor completely, and is very useful geometric notion. In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

Riemannian manifolds with constant sectional curvature are the most simple. These are called space forms. By rescaling the metric there are three possible cases In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a space form is a complete Riemannian manifold M of constant curvature, K. It is a theorem of Riemannian geometry that the universal cover of an n dimensional space form with curvature K = -1 is isometric to , hyperbolic space, with curvature K = 0 is isometric to , Euclidean n...

• 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 . A triangle immersed in a saddle-shape plane (an hyperbolic paraboloid), as well as two diverging parallel lines. ... Euclid Euclidean geometry is a mathematical system due to the Hellenistic mathematician Euclid of Egypt. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... For other uses, see sphere (disambiguation). ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...

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.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

See also

• Riemann curvature tensor
• curvature of Riemannian manifolds
• curvature