pair, Codazzi pair, and some of their associated invariants such as the mean curvature $H$, the extrinsic curvature $K$ and the Hopf differential. We prove in Theorem \[elteorema\] that a topological sphere $\Sigma$ endowed with a Codazzi pair satisfying a general Weingarten relationship $W(H,K)=0$ must be totally umbilical, if some necessary conditions are fulfilled by the functional $W$. This generalizes the previous Hopf theorem and Liebmann theorem.

This abstract treatment lets us apply Theorem \[elteorema\] to some seemingly unrelated situations. More specifically, as a consequence of that result, we obtain generalizations of the Bonnet Theorem and the theorem of uniqueness in the Christoffel problem. We point out that our proof to the Christoffel problem is different from the classical approach, which uses integration theory on surfaces (see [@H; @Sp]).

We will finish Section \[s2\] proving that two Codazzi pairs $(I_i,II_i)$, $i = 1,2$, on a topological sphere $\Sigma$, such that $II_1=II_2$ and with the same positive extrinsic curvature must be isometric, that is, $I_1=I_2$. This result is a wide generalization of a classical result by Grove [@Gr] about rigidity of ovaloids in $\r^3$.

In Section \[s3\] we study when a real quadratic form $II$ on a Riemannian surface is conformal 