[@AdCT Main Lemma] (see also [@Jost Lemma 2.7.1]), either $Q$ vanishes identically on $U$ or $p$ is an isolated zero of negative index of $Q$.

In particular, if ${\Sigma}$ is a topological sphere, from the Poincaré index Theorem we get that the Hopf differential $Q\,dz^2$ must vanish identically on ${\Sigma}$, as we wanted to prove.

The above result can be globally used not only for topological spheres. Indeed, if ${\Sigma}$ is a topological torus under the assumptions of Theorem \[elteorema\], then we deduce that the pair $(I,II)$ is either totally umbilical or umbilically free. Analogously, if ${\Sigma}$ is a closed topological disk and its boundary $\partial{\Sigma}$ is a line of curvature for $(I,II)$, then the pair is totally umbilical.

It is well-known that the hypothesis (\[hipotesis\]) cannot be removed. Examples of this are the non totally umbilical rotational spheres in any space form, since every rotational sphere is a Weingarten surface.

The abstract use of Codazzi pairs allows us to see some classical results, apparently non related, as immediate consequences of Theorem \[elteorema\]. Two good examples are Bonnet Theorem and the uniqueness to the Christoffel problem in $\r^3$, as we show next

\[bonnet\] [**(Abstract Bonnet Theorem)**]{} Let ${\Sigma}$ be a 