}{\partial z}\right)+I_i\left(S_i{\frac{\partial\ }{\partial
z}},\nabla^i_{\frac{\partial\ }{\partial z}}\frac{\partial\
}{\partial {\bar{z}}}\right)\end{aligned}$$ and so $$\label{formula1}
\rho_z=\rho\left(\Gamma_{11}^{1,i}-\overline{\Gamma_{12}^{1,i}}\right).$$

On the other hand, a direct calculation gives $$\left(\lambda_i^2-|P_i|^2\right)_z=2\,\left(\lambda_i^2-|P_i|^2\right)\,\left(\Gamma_{11}^{1,i}+
\overline{\Gamma_{12}^{1,i}}\right).$$

With all of this, we obtain from (\[HK\]) by differentiating $K$ $$\label{formula2}
\frac{K_z}{K}=2\frac{\rho_z}{\rho}-2\left(\Gamma_{11}^{1,i}+
\overline{\Gamma_{12}^{1,i}}\right).$$ Therefore, from (\[formula1\]) and (\[formula2\]), $$\frac{K_z}{K}=-4\,\overline{\Gamma_{12}^{1,i}}=\frac{2K}{\rho^2}\left(P_i\,\overline{P_i}_z-\lambda_i\,{P_i}_{{\bar{z}}}\right),$$ or equivalently, $$\label{f1}
{P_i}_{{\bar{z}}}=-\frac{1}{2K}\left(\lambda_i\,K_z+P_i\,K_{{\bar{z}}}\right).$$

Moreover, from (\[HK\]) we get $$\begin{aligned}
|\lambda_1-\lambda_2|&=&\frac{\rho}{K}\,|H_1-H_2|=\frac{\rho}{K}\,\sqrt{(H_1-H_2)^2} \label{f2} \\
&\leq& \frac{\rho}{K}\,\left|\sqrt{H_1^2-K}-\sqrt{H_2^2-K}\right|=
\left||P_1|-|P_2|\right| \leq |P_1-P_2|. \nonumber\end{aligned}$$

Now, we can derive from (\[f1\]) and (\[f2\]) that $$|{P_1}_{{\bar{z}}}-{P_2}_{{\bar{z}}}|\leq\frac{|K_z|+|K_{{\bar{z}}}|}{2K}\,|P_1-P_2|.$$ Finally, using [@AdCT Main Lemma] or [@Jost Lemma 2.7.1], we conclude that the quadratic form $(P_1-P_2)dz^2$ vanishes identically on the topological sphere ${\Sigma}$. Thus, $P_1\equiv P_2$ and, from (\[HK\]), we get $\lambda_1=\lambda_2$. Or equivalently, $I_1=I_2$.

The Codazzi function {#s3}
====================

Under certain natural conditions, it is possible to obtain important consequences about a surface endowed with a fundamental pair although the Codazzi equation is not satisfied. In order to study these conditions, next we define the Codazzi tensor and the Codazzi function, which will play an essential role in our study.

\[d4\] Given a fundamental pair $(I,II)$ on a surface ${\Sigma}$ with associated shape operator $S$, we will call Codazzi tensor of $(I,II)$ to the map $T_S:{\mathfrak{X}}({\Sigma})\times {\mathfrak{X}}({\Sigma})\fl {\mathfrak{X}}({\Sigma})$ defined by $$T_S(X,Y)=\nabla_XSY-\nabla_YSX-S[X,Y],\qquad X,Y\in {\mathfrak{X}}({\Sigma}).$$

Although the definition above has 