Gaussian curvature - Revision history

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2008-05-18T19:41:19Z

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2008-04-06T14:31:51Z

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	The equivalence of these definitions arises from Gauss's [[Theorema Egregium]].		The equivalence of these definitions arises from Gauss's [[Theorema Egregium]].

			==Generalizations==

			===Generalization of the abstract definition===

			{{further\|[[sectional curvature]],[[Ricci curvature]],[[scalar curvature]]}}

			The abstract definition generalizes to the notion of [[sectional curvature]]. Sectional curvature is a ''two-dimensional'' curvature on manifolds of higher dimension: it takes as input a point on a manifold and a tangent plane at the point, it outputs the [[Gaussian curvature]] that we'd get by taking a two-dimensional submanifold with that tangent plane as tangent space.

			The [[Ricci curvature]] takes a tangent line instead of a tangent plane, and gives the ''average'' sectional curvature over all tangent planes containing that tangent line. The scalar curvature averages over all tangent lines.

			===Generalization of the concrete definition===

			{{further\|[[Gauss-Kronecker curvature]]}}

			The Gauss-Kronecker curvature is valid for codimension one submanifolds (or hypersurfaces) in Euclidean space of any dimension, and uses precisely the same definition as that of the Gaussian curvature.