The equations of Gauss, Codazzi and Ricci of surfaces in 4-dimensional space forms

Let $N$ be a Riemannian or neutral $4$-dimensional space form. In this paper, the expressions of the equations of Gauss, Codazzi and Ricci of a space-like or time-like surface in $N$ given in [6] are naturally understood in terms of the induced connection of the two-fold exterior power of the pull-back bundle on the surface. We observe that a space-like or time-like surface in $N$ such that the twistor lifts are nondegenerate is given by relations among functions in the above expressions related to the second fundamental form. In the case where $N$ is a Lorentzian $4$-dimensional space form, we define the complex twistor lifts of a space-like or time-like surface in $N$ and we have analogous discussions and results in terms of the induced connection of the complexification of the two-fold exterior power of the pull-back bundle. We characterize a space-like surface in $N$ such that the covariant derivative of a suitable complex twistor lift by $\partial /\partial \overline{w}$ vanishes.