Generic conformally flat hypersurfaces and surfaces in 3-sphere

Abstract

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere.The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition.Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition.In this paper, we firstly relate 2-metrics of the class to surfaces in the 3-sphere:for a 2-metric of the class, a 5-dimensional set of (non-isometric) analytic surfaces in the 3-sphere is determined such thatany surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surfaceand the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces.