We study generic conformally flat hypersurfaces in
the Euclidean $4$-space. The conformal flatness condition for the
Riemannian metric is given by a set of several differential equations of
order
three. In this paper, we first define a certain class $\Xi$ of metrics for
3-manifolds which includes, as a large subset, all metrics of generic
conformally flat hypersurfaces in the Euclidean 4-space.
We obtain a kind of integrability condition on metrics of the class.
Restricting our consideration to metrics of conformally
flat hypersurfaces, we define a conformal invariant for generic
conformally flat hypersurfaces and obtain
a differential equation of order three from the integrability condition.
The equation is equal to the simplest one in equations of conformal
flatness condition.
Next, we study some particular solutions of the equation. We will
determine all
generic conformally flat hyersurfaces corresponding
to these particular solutions under an assumption on
the first fundamental form, and characterize these hypersurfaces
geometrically.
The result includes
all known examples of generic conformally flat hypersurfaces in
the Euclidean $4$-space.
All known examples are the following:
The hypersurfaces given by Lafontaine ([6]) which are made from constant
Gaussian curvature surface in the three dimentional space forms, the hypersurfaces given by
Suyama ([7]), and the flat metrics obtained by Hertrich-Jeromin ([3]).
Furthermore, we explicitly construct a series of
examples of generic conformally flat hypersurface, which have a
geometrical property different from all known examples.
Then, we have the following case: There exists a pair of hypersurfaces
with the same conformal invariant, each of which is constructed from a
surface with constant Gaussian curvature in either the Euclidean 3-space or
the standard 3-sphere but does not belong to the known examples.
Furthermore, no conformal transformation
maps diffeomorphically one hypersurface of the pair to the other hypersurface.