$S$, respectively.

In particular, given local parameters $(x,y)$ on ${\Sigma}$ such that $$I=E\,dx^2+2F\,dxdy+G\,dy^2,\qquad II=e\,dx^2+2f\,dxdy+g\,dy^2,$$ the mean curvature and the extrinsic curvature of the pair are given, respectively, by $$H=H(I,II)=\frac{E g+G e-2F f}{2(EG-F^2)},\qquad K=K(I,II)=\frac{eg-f^2}{EG-F^2}.$$ Moreover, the principal curvatures of the pair are $H\pm\sqrt{H^2-K}$.

We will say that the pair $(I,II)$ is [*umbilical*]{} at $p\in {\Sigma}$ if $II$ is proportional to $I$ at $p$, or equivalently:

-   if both principal curvatures coincide at $p$, or

-   if $S$ is proportional to the identity map on the tangent plane at $p$, or

-   if $H^2-K=0$ at $p$.

We define the [*Hopf differential*]{} of the fundamental pair $(I,II)$ as the (2,0) part of $II$ for the Riemannian metric $I$. In other words, if we consider $\Sigma$ as a Riemann surface with respect to the metric $I$ and take $z$ a local conformal parameter, then $$\label{parholomorfo}
\begin{array}
{c} I=2\lambda\,|dz|^2\\[2mm]
II=Q\,dz^2+2\lambda\,H\,|dz|^2+\overline{Q}\,d{\bar{z}}^2.
\end{array}$$ The quadratic form $Q\,dz^2$, which does not depend on the chosen parameter, is known as the Hopf differential of the pair $(I,II)$. We note that $(I,II)$ is umbilical at $p\in\Sigma$ if, and only if, $Q(p)=0$.

All the above definitions can be understood as natural extensions of the corresponding ones for isometric immersions of a 