Gauss-Kronecker curvature of a hypersurface

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Definition

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

K:M \to \R

defined in the following equivalent ways:

Particular cases

When n = 2, we get the usual notion of Gaussian curvature.

Facts