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

Let $N$ be a Riemannian, neutral or Lorentzian $4$-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 [7] are naturally understood in terms of the induced connection (of the complexification) of the two-fold exterior power of the pull-back bundle on the surface. Moreover, based on such expressions, we characterize several classes of surfaces related to the covariant derivatives of the twistor lifts and so on.