Gauss-Kronecker curvature of a hypersurface

Definition

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

$K:M\to \mathbb{R}$

defined in the following equivalent ways:

• $K$ is the Jacobian determinant of the Gauss map of $M$
• $K$ is the determinant of the shape operator for $M$

Particular cases

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

Facts

• Gauss-Bonnet theorem for hypersurfaces