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2008-05-18T19:41:14Z

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Vipul: New page: ==Definition== Let $M$ be a hypersurface in $\R^{n+1}$ : in other words $M$ is a $n$ -dimensional manifold embedded inside $\R^{n+1}$ ....

2008-04-06T14:25:36Z

New page: ==Definition== Let <math>M</math> be a hypersurface in <math>\R^{n+1}</math>: in other words <math>M</math> is a <math>n</math>-dimensional manifold embedded inside <math>\R^{n+1}</math>....

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==Definition==

Let <math>M</math> be a hypersurface in <math>\R^{n+1}</math>: in other words <math>M</math> is a <math>n</math>-dimensional manifold embedded inside <math>\R^{n+1}</math>. The '''Gauss-Kronecker curvature''' of <math>M</math> is a function:

<math>K:M \to \R</math>

defined in the following equivalent ways:

* <math>K</math> is the Jacobian determinant of the [[Gauss map]] of <math>M</math>
* <math>K</math> is the determinant of the [[shape operator]] for <math>M</math>

==Particular cases==

When <math>n = 2</math>, we get the usual notion of [[Gaussian curvature]].

==Facts==

* [[Gauss-Bonnet theorem for hypersurfaces]]

Gauss-Kronecker curvature of a hypersurface - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== Let M be a hypersurface in \R^{n+1}: in other words M is a n-dimensional manifold embedded inside \R^{n+1}....

Vipul: New page: ==Definition== Let $M$ be a hypersurface in $\R^{n+1}$ : in other words $M$ is a $n$ -dimensional manifold embedded inside $\R^{n+1}$ ....