sectional curvature

Let $M$ be a Riemannian manifold. Let $p$ be a point in $M$ and let $S$ be a two-dimensional subspace of $T_{p}M$. Then the sectional curvature of $S$ at $p$ is defined as

$$K(S)=\frac{g(R(x,y)x,y)}{g(x,x)g(y,y)-g{(x,y)}^2}$$

where $x,y$ span $S$, $g$ is the metric tensor and $R$ is the Riemann’s curvature tensor.

This is a natural generalization of the classical Gaussian curvature for surfaces.