Define the terms Gaussian curvature $K$ and mean curvature $H$ for a smooth embedded oriented surface $S \subset \mathbf{R}^{3}$. [You may assume the fact that the derivative of the Gauss map is self-adjoint.] If $K=H^{2}$ at all points of $S$, show that both $H$ and $K$ are locally constant. [Hint: Use the symmetry of second partial derivatives of the field of unit normal vectors.]

If $K=H^{2}=0$ at all points of $S$, show that the unit normal vector $\mathbf{N}$ to $S$ is locally constant and that $S$ is locally contained in a plane. If $K=H^{2}$ is a strictly positive constant on $S$ and $\phi: U \rightarrow S$ is a local parametrization (where $U$ is connected) on $S$ with unit normal vector $\mathbf{N}(u, v)$ for $(u, v) \in U$, show that $\phi(u, v)+\mathbf{N}(u, v) / H$ is constant on $U$. Deduce that $S$ is locally contained in a sphere of radius $1 /|H|$.

If $S$ is connected with $K=H^{2}$ at all points of $S$, deduce that $S$ is contained in either a plane or a sphere.