Shape operator on a hypersurface
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 .