as an analytic tool to derive new global results for surfaces in the space forms $\r^3$, $\s^3$ and $\h^3$.

Our results on Codazzi pairs here provide an extremely general extension of some classical theorems of surface theory that mainly depend on the Codazzi equation. But, moreover, this abstract approach has some very definite applications to the study of complete or properly embedded Weingarten surfaces in $\r^3$ or $\h^3$:

1.  It reveals the existence of holomorphic quadratic differentials for some classes of surfaces in space forms (and also in product spaces $\s^2\times \r$, $\h^2\times \r$, see [@AEG1]).

2.  It unifies the proof of apparently non related theorems. For example, it shows that uniqueness in the Christoffel problem in $\r^3$ is basically equivalent to the Bonnet theorem on uniqueness of immersed spheres with prescribed mean curvature.

3.  It gives an analytic tool to prove uniqueness results for complete or compact Weingarten surfaces in space forms.

These applications show the flexibility of the use of Codazzi pairs in surface theory, and suggest the possibility of obtaining further global results with the techniques employed here.

We have organized this paper as follows. We shall start by reminding in Section \[s2\] the definitions of fundamental 