Shape Operator

Definition

Linear operator on tangent space $S_p:T_{p}S\to T_{p}S$. If $\mathbf{n}$ is a unit normal vector to $S$ and $\mathbf{v}$ is a tangent vector then $S_p(\mathbf{v})=-{\nabla }_{\mathbf{v}}\mathbf{n}$

Facts

The Gaussian Curvature $K=\det (S)$ where $S$ is a Shape Operator

The Mean Curvature $K=\frac{1}{2}\mathrm{tr}(S)$ where $S$ is a Shape Operator