shape operator

The shape operator $S$ of a surface $\Sigma$ in ${\mathbb{R}}^{3}$ is the derivative of the sphere map $N:\Sigma\to S^{2}$ given by $N(p)=$ unit normal vector field at $p$ . So at each $p$ , $S(p)=d_{p}N$ and it is the linear transformation $S(p):T_{p}\Sigma\to T_{N(p)}S^{2}$ . This is important, because the determinant defines the Gaussian curvature at $p$ in $\Sigma$ .