Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ${\displaystyle K(\sigma _{p})}$ depends on a two-dimensional linear subspace ${\displaystyle \sigma _{p}}$ of the tangent space at a point ${\displaystyle p}$ of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane ${\displaystyle \sigma _{p}}$ as a tangent plane at ${\displaystyle p}$, obtained from geodesics which start at ${\displaystyle p}$ in the directions of ${\displaystyle \sigma _{p}}$ (in other words, the image of ${\displaystyle \sigma _{p}}$ under the exponential map at ${\displaystyle p}$). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

The sectional curvature determines the Riemann curvature tensor completely.