= Solution
Choose a smooth <unit normal> $N$. For $v,w\in T_p\Sigma$, the <first fundamental form> and <second fundamental form> are
$$
I_p(v,w)=\langle v,w\rangle,
\qquad
II_p(v,w)=\langle S_pv,w\rangle,
$$
where the shape operator is $S_p=-dN_p$. The principal curvatures are the <eigenvalues> $\kappa_1,\kappa_2$ of $S_p$; the <mean curvature> and <Gaussian curvature> are
$$
H=\frac12\operatorname{tr}S_p=\frac{\kappa_1+\kappa_2}{2},
\qquad
K=\det S_p=\kappa_1\kappa_2.
$$
The surface is minimal when $H=0$ everywhere.
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