Shape operator

Summary

The shape operator of a surface in $\R^3$ at a point is a linear map of the tangent plane to itself. It tells how the unit normal turns as one moves on the surface. It is minus the differential of the Gauss map, and it is also called the Weingarten map. Its eigenvalues are the principal curvatures, its determinant is the Gaussian curvature, and half its trace is the mean curvature.