Sectional curvature

Summary

The sectional curvature of a Riemannian manifold assigns a number to each $2$-dimensional plane in each tangent space. It is the Gaussian curvature at $p$ of the surface swept out by the geodesics from $p$ tangent to that plane. The sectional curvatures determine the full Riemann curvature tensor, and bounds on them are the hypotheses of the main comparison theorems of global Riemannian geometry.