Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional plane σp in the tangent space at a point p of the manifold. 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. In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional plane σp in the tangent space at a point p of the manifold. 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. Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define Here R is the Riemann curvature tensor. If we take the opposite sign for the curvature tensor, then we should reverse the sign of the previous quotient so the sphere has positive sectional curvature. In particular, if u and v are orthonormal, then The sectional curvature in fact depends only on the 2-plane σp in the tangent space at p spanned by u and v. It is called the sectional curvature of the 2-plane σp, and is denoted K(σp). Riemannian manifolds with constant sectional curvature are the simplest. These are called space forms. By rescaling the metric there are three possible cases The model manifolds for the three geometries are hyperbolic space, Euclidean space and a unit sphere. They are the only connected, complete, simply connected Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of isometries.