Dual pairs of generic conformally flat hypersurfaces

Abstract

We study generic conformally flat (local-)hypersurfaces in the Euclidean 4-space $\mathbb{R}^4$. Such a hypersurface $f$ has the dual (hypersurface) $f^*$ in $\mathbb{R}^4$, which is also generic and conformally flat. By repeating the composite action of inversion and the dual transformation on a hypersurface $f$, infinitely many non-equivalent generic conformally flat hypersurfaces are obtained from a single $f$. The dual $f^*$ is defined by the total differential of the embedding expressed in terms of the original $f$. However, the exact formula in $\mathbb{R}^4$ of $f^*$ is not obvious, because of difficulty of integrating the total differential. Therefore, for the study of generic conformally flat hypersurfaces it is important to clarify in some way an explicit correspondence between the dual pair in $\mathbb{R}^4$. The aim of this paper is to clarify that correspondence between the dual pair $f$ and $f^*$, which we do by making approximate discrete hypersurfaces of $f^*$ for all positive integers $n$ as maps from $3$-dimensional nets in $f$ to $\mathbb{R}^4$. The approximations are constructed from the dual invariants of a generic conformally flat hypersurface $f$, of which dual invariants are defined as the maps from $f$ to $\mathbb{R}^4$, and as $n$ tends to $\infty$, the approximations induce maps between the corresponding curvature surfaces of $f$ and $f^*$.