  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 that any surface of the set gives rise to an
evolution of surfaces in the 3-sphere issuing from the surface and 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.
