for a fundamental pair $(I,II)$, then we can write $$I=E\,du^2+G\,dv^2,\qquad II=k_1\,E\,du^2+k_2\,G\,dv^2.$$ Hence, the Codazzi tensor acting on the vector fields $\frac{\partial\ }{\partial u},\frac{\partial\ }{\partial v}$ can be expressed as $$\begin{aligned}
T_S\left(\frac{\partial\ }{\partial u},\frac{\partial\ }{\partial
v}\right)&=&\nabla_{\frac{\partial\ }{\partial u}}S\frac{\partial\ }{\partial
v}-\nabla_{\frac{\partial\ }{\partial v}}S\frac{\partial\ }{\partial
u}\,=\,\nabla_{\frac{\partial\ }{\partial u}}k_2\frac{\partial\ }{\partial
v}-\nabla_{\frac{\partial\ }{\partial v}}k_1\frac{\partial\ }{\partial u}\nonumber\\
&=&(k_2)_u\,\frac{\partial\ }{\partial v}-(k_1)_v\,\frac{\partial\ }{\partial
u}+(k_2-k_1)\left(\frac{E_v}{2E}\frac{\partial\ }{\partial
u}+\frac{G_u}{2G}\frac{\partial\ }{\partial v}\right)\label{Tuv}\\
&=&-\frac{1}{E}\left((k_1E)_v-H\,E_v\right)\,\frac{\partial\ }{\partial u}+
\frac{1}{G}\left((k_2G)_u-H\,G_u\right)\,\frac{\partial\ }{\partial v}\nonumber\,,\end{aligned}$$ where $H$ is the mean curvature of $(I,II)$.

We observe that we can take doubly orthogonal parameters in a neighborhood of every non umbilical point as well as in a neighborhood of every point in the interior of ${\Sigma}_U=\{\mbox{umbilical points of }(I,II)\}$. Thus, the set of points where there exist doubly orthogonal parameters is dense in ${\Sigma}$. Consequently, all the properties that we prove by using this kind of parameters, will be extended to the whole surface by continuity.

Throughout this section we will use the new quadratic form $II'$ associated to the fundamental pair $(I,II)$ given by $$II'=II-H\,I.$$

\[lemita\] Let $(I,II)$ be a Codazzi pair on a surface ${\Sigma}$ with mean and extrinsic curvatures $H$ and $K$ respectively. Let $\varphi$ be a differentiable function on ${\Sigma}$ such that the function $\sinh\varphi/\sqrt{H^2-K}$ can be 