Conformally flat hypersurfaces in Euclidean 4-space II

Abstract

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.