PlanetMath: sectional curvature

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sectional curvature

(Definition)

Let be a Riemannian manifold. Let be a point in and let be a two-dimensional subspace of . Then the sectional curvature of at is defined as

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

where

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

"sectional curvature" is owned by juanman.

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Keywords:

curvature

Attachments:: sectional curvature determines Riemann curvature tensor (Theorem) by kerwinhui

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Cross-references: surfaces, Gaussian curvature, Riemann's curvature tensor, metric tensor, span, subspace, point, Riemannian manifold
There are 4 references to this entry.

This is version 2 of sectional curvature, born on 2006-05-08, modified 2006-09-07.
Object id is 7907, canonical name is SectionalCurvature.
Accessed 1930 times total.

Classification:

AMS MSC:	53B20 (Differential geometry :: Local differential geometry :: Local Riemannian geometry)
	53B21 (Differential geometry :: Local differential geometry :: Methods of Riemannian geometry)

Pending Errata and Addenda

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