%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: sectional curvature
%%% Primary Category Code: 53B20
%%% Filename: SectionalCurvature.tex
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%%% Owner: juanman
%%% Author(s): juanman
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\begin{document}

 Let $M$ be a \htmladdnormallink{Riemannian manifold}{http://planetmath.org/encyclopedia/RiemannianStructure.html}. Let $p$ be a \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} in $M$ and let $S$ be a two-dimensional \htmladdnormallink{subspace}{http://planetmath.org/encyclopedia/ProperVectorSubspace.html} 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 $x,y$ \htmladdnormallink{span}{http://planetmath.org/encyclopedia/SpanningSet.html} $S$, $g$ is the \htmladdnormallink{metric tensor}{http://planetmath.org/encyclopedia/RiemannianStructure.html} and $R$ is the \htmladdnormallink{Riemann's curvature tensor}{http://planetmath.org/encyclopedia/RiemannCurvatureTensor.html}.

This is a natural generalization of the classical \htmladdnormallink{Gaussian curvature}{http://planetmath.org/encyclopedia/GaussianCurvature.html} for \htmladdnormallink{surfaces}{http://planetmath.org/encyclopedia/Surface.html}.

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