The equations of Gauss, Codazzi and Ricci of surfaces in 4-dimensional space forms
Naoya Ando
TL;DR
This work unifies the treatment of Gauss–Codazzi–Ricci equations for space-like and time-like surfaces in four-dimensional space forms across Riemannian, neutral, and Lorentzian geometries by interpreting them through the induced connections on the wedge^2 pull-back bundle. It develops a comprehensive twistor-theoretic framework, including real and complex twistor lifts, and identifies precise degeneracy and nondegeneracy criteria (notably in terms of Δ and related data) that determine when a surface is determined by intrinsic data up to ambient isometries. In the Lorentzian setting, it introduces a complexification of wedge^2 to construct complex twistor spaces and analyzes vanishing along ∂̄ of twistor lifts, linking geometric properties to holomorphic data and isotropy. The results provide a systematic method to reconstruct immersed surfaces from scalar data (λ, α_k, β_k, μ_l) subject to Gauss–Codazzi–Ricci equations, and to classify special isotropic/zero-mean-curvature cases via twistor degeneration and holomorphic differentials.
Abstract
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.