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

TL;DR

This paper derives and interprets the Gauss, Codazzi, and Ricci equations for surfaces in 4D space forms, linking them to complexified connections and characterizing related surface classes.

Contribution

It provides a natural understanding of these fundamental equations in terms of complexified connections and characterizes classes of surfaces via twistor lifts.

Findings

Expressions of Gauss, Codazzi, Ricci equations are given in terms of induced complexified connections.

Characterization of surface classes related to covariant derivatives of twistor lifts.

Unified interpretation of surface equations in 4D space forms.

Abstract

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.