# Gauss-Kronecker curvature of a hypersurface

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## Definition

Let be a hypersurface in : in other words is a -dimensional manifold embedded inside . The **Gauss-Kronecker curvature** of is a function:

defined in the following equivalent ways:

- is the Jacobian determinant of the Gauss map of
- is the determinant of the shape operator for

## Particular cases

When , we get the usual notion of Gaussian curvature.