# Shape operator on a hypersurface

From Diffgeom

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## Definition

### For a hypersurface in any dimension

Suppose is a -dimensional manifold embedded inside . The **shape operator** on associates, to every point , a linear map from to , given by:

where is the component of (the covariant derivative of the normal in terms of ) in the -direction.

Equivalently, the shape operator is the differential of the Gauss map for the hypersurface , namely the map:

that sends a point in to the normal direction at that point.

The shape operator can be viewed as a section of the bundle .

### For a regular surface in 3-space

This is the special case of the above, in the situation where is a regular surface inside .