sectional curvature SectionalCurvature 2006-05-08 14:58:19 2006-09-07 09:51:01 Definition curvature % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $M$ be a Riemannian manifold. Let $p$ be a point in $M$ and let $S$ be a two-dimensional subspace of $T_pM$. Then the \emph{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.