Riemann surface in a 3-dimensional ambient space, where $I$ plays the role of the induced metric and $II$ the role of its second fundamental form.

A specially interesting case happens when the fundamental pair satisfies, in an abstract way, the Codazzi equation for surfaces in $\r^3$,

We say that a fundamental pair $(I,II)$, with shape operator $S$, is a Codazzi pair if $$\label{ecuacioncodazzi}
\nabla_XSY-\nabla_YSX-S[X,Y]=0,\qquad X,Y\in{\mathfrak{X}}({\Sigma}),$$ where $\nabla$ stands for the Levi-Civita connection associated to the Riemannian metric $I$ and ${\mathfrak{X}}({\Sigma})$ is the set of differentiable vector fields on ${\Sigma}$.

Many Codazzi pairs appear in a natural way in the study of surfaces. For instance, the first and second fundamental forms of a surface isometrically immersed in a 3-dimensional space form is a Codazzi pair. The same occurs for spacelike surfaces in a 3-dimensional Lorentzian space form. More generally, if the surface is immersed in an $n$-dimensional (semi-Riemannian) space form and has a parallel unit normal vector field $N$, then its induced metric and its second fundamental form associated to $N$ make up a Codazzi pair.

Classically, Codazzi pairs also arise in the study of harmonic maps. Many others examples of Codazzi pairs also appear in [@AEG1; @Bi; @Mi; @Ol]. All 