second fundamental tensor | Weingarten map

The negative derivative S(v) = - D_v N of the unit normal N vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The shape operator S is an extrinsic curvature, and the Gaussian curvature is given by the determinant of S. If x:U->R^3 is a regular patch, then S(x_u) | = | -N_u S(x_v) | = | -N_v. At each point p on a regular surface M subset R^3, the shape operator is a linear map S:M_p->M_p.

curvature | fundamental forms | Weingarten equations