# 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.

Title sectional curvature SectionalCurvature 2013-03-22 15:54:15 2013-03-22 15:54:15 juanman (12619) juanman (12619) 5 juanman (12619) Definition msc 53B21 msc 53B20 RiemannianMetric