shape operator

The shape operator $S$ of a surface $\mathrm{\Sigma }$ in ${ℝ}^3$ is the derivative of the sphere map $N:\mathrm{\Sigma }\to S^2$ 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 $S(p):T_p\mathrm{\Sigma }\to T_{N(p)}{S}^2$. This is important, because the determinant defines the Gaussian curvature at $p$ in $\mathrm{\Sigma }$.