$II(X,Y)=I(S(X),Y)$, where $II$ is the second fundamental form of the surface. This Codazzi equation is, together with the Gauss equation, one of the two classical integrability conditions for surfaces in $\r^3$, and it remains invariant if we substitute the ambient space $\r^3$ by other space form $\s^3$ or $\h^3$.

It is remarkable that some crucial results of surface theory in $\r^3$ only depend, in essence, of the Codazzi equation. This is the case, for instance, of Hopf’s theorem (resp. Liebmann’s theorem) on the uniqueness of round spheres among immersed constant mean curvature spheres (resp. among complete surfaces of constant positive curvature). This suggests the possibility of adapting these results to an abstract setting of *Codazzi pairs* (i.e. pairs of real quadratic forms $(I,II)$ on a surface verifying ), and to explore its possible consequences in surface theory. The basic idea in this sense is to use the Codazzi pair $(I,II)$ as a geometric object in a *non-standard* way, i.e. so that $(I,II)$ are no longer the first and second fundamental forms of a surface in a space form.

Motivated by this, our objective here is to develop an abstract theory for the Codazzi equation on surfaces, and use it subsequently 