of this shows that the results that we present in this work can be used in many different contexts.

Many classical results in surface theory depend on the Codazzi equation of the immersion. This fact allows to generalize such results to the Codazzi pairs theory. Some examples of that, are Hopf’s results proving that the only surfaces immersed in $\r^3$ with constant mean curvature are totally umbilical. Analogously, Liebmann proved that the only complete surfaces with positive constant Gaussian curvature in $\r^3$ are totally umbilical spheres. Now, we obtain a generalization of both results to the wider family of Weingarten pairs, which we define next

We say that a fundamental pair $(I,II)$ on a surface ${\Sigma}$ is a Weingarten pair if its mean and extrinsic curvatures, $H$ and $K$ respectively, satisfy a non trivial relationship $$W(H,K)=0,$$ where $W$ is a differentiable function defined on an open set of $\r^2$ containing the set of points $\{(H(p),K(p)):\ p\in{\Sigma}\}$.

\[elteorema\] Let $(I,II)$ be a Codazzi pair on a surface ${\Sigma}$. If $(I,II)$ is a Weingarten pair for a functional $W(x,y)$ such that $$\label{hipotesis} W_x(t,t^2)+2t\,W_y(t,t^2)\neq0\qquad\mbox{for all
} t,$$ then either the umbilical points of $(I,II)$ are isolated and of negative index, or the pair 