been made in an abstract context, the Codazzi tensor appears naturally in the study of isometric immersions of surfaces. To be more precise, the Codazzi equation of a surface isometrically immersed in a 3-dimensional manifold $M^3$ is $$\nabla_XSY-\nabla_YSX-S[X,Y]=-\overline{R}(X,Y)N,\qquad X,Y\in{\mathfrak{X}}({\Sigma}),$$ where $N$ is the unit normal vector field of the immersion, $S$ the associated shape operator and $\overline{R}$ the curvature tensor of $M^3$ $$\overline{R}(X,Y)Z=\overline{\nabla}_X\overline{\nabla}_YZ-
\overline{\nabla}_Y\overline{\nabla}_XZ-\overline{\nabla}_{[X,Y]}Z,$$ $\overline{\nabla}$ being the Levi-Civita connection of $M^3$.

A straightforward computation shows that the Codazzi tensor of a fundamental pair on a surface satisfies the following properties:

\[l1\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape operator $S$ and Codazzi tensor $T_S$. Then

1.  $T_S$ is skew-symmetric, i.e. $T_S(X,Y)=-T_S(Y,X)$ for all $X,Y\in {\mathfrak{X}}({\Sigma})$.

2.  $T_S$ is ${\cal C}^{\infty}(\Sigma)$-bilineal, that is, $$T_S(f_1\,X_1+f_2\,X_2,Y)=f_1\,T_S(X_1,Y)+f_2\,T_S(X_2,Y)$$ for all vector fields $X_1,X_2,Y\in {\mathfrak{X}}({\Sigma})$ and differentiable real functions $f_1,f_2$.

3.  Moreover, given vector fields $X,Y\in {\mathfrak{X}}({\Sigma})$ and a differentiable real function $f$ on ${\Sigma}$ $$T_{fS}(X,Y)=f\,T_S(X,Y)+X(f)\,SY-Y(f)\,SX.$$

Associated to the Codazzi tensor of a fundamental pair we define the Codazzi function, thanks to which we will [*measure how distant*]{} the pair is from satisfying the Codazzi equation.

\[d5\] Let $(I,II)$ be a fundamental pair on a surface ${\Sigma}$ with associated shape 