Shape Operator

The negative derivative

$$S({\bf v})=-D_{\bf v}{\bf N}$$ (1)

of the unit normal ${\bf N}$ 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 ${\bf x}:U\to\Bbb{R}^3$ is a Regular Patch, then

$$S({\bf x}_u) = -{\bf N}_u$$ (2)

$$S({\bf x}_v) = -{\bf N}_v.$$ (3)

At each point p on a Regular Surface $M\subset\Bbb{R}^3$, the shape operator is a linear map

$$S:M_{\bf p}\to M_{\bf p}.$$ (4)

The shape operator for a surface is given by the Weingarten Equations.

See also Curvature, Fundamental Forms, Weingarten Equations

References

Gray, A. ``The Shape Operator,'' ``Calculation of the Shape Operator,'' and ``The Eigenvalues of the Shape Operator.'' §14.1, 14.3, and 14.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 268-269, 274-279, 1993.

Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 30, 1986.