it is a right circular cylinder or a surface of minimal type (i.e. $f(0)=0$).

Fundamental pairs and Codazzi pairs {#s2}
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Let us start this section by recalling some classical results about fundamental pairs. A classical reference about this topic is [@Mi]. Besides we point out that, although throughout this paper we will assume that the differentiability used is always $\cal{C}^{\infty}$, the differentiability requirements are much lower.

We will denote by ${\Sigma}$ an orientable (and oriented) differentiable surface. Otherwise we would work with its oriented two-sheeted covering.

A fundamental pair on ${\Sigma}$ is a pair of real quadratic forms $(I,II)$ on ${\Sigma}$, where $I$ is a Riemannian metric.

Associated to a fundamental pair $(I,II)$ we define the shape operator $S$ of the pair as $$\label{ii}
 II(X,Y)=I(S(X),Y)$$ for any vector fields $X,Y$ on ${\Sigma}$.

Conversely, from (\[ii\]) it becomes clear that the quadratic form $II$ is totally determined by $I$ and $S$. In other words, giving a fundamental pair on ${\Sigma}$ is equivalent to giving a Riemannian metric on ${\Sigma}$ and a self-adjoint endomorphism $S$.

We define the [*mean curvature*]{}, the [*extrinsic curvature*]{} and the [*principal curvatures*]{} of $(I,II)$ as half the trace, the determinant and the eigenvalues of the endomorphism 