A CLASSIFICATION AND A NON-EXISTENCE THEOREM FOR CONFORMALLY FLAT HYPERSURFACES IN EUCLIDEAN 4-SPACE

Abstract

We study generic conformally flat hypersurfaces in the Euclidean 4-space satisfying a certain condition on the conformal class of the first fundamental form. We first classify such hypersurfaces by determining all conformal-equivalence classes of generic conformally flat hypersurfaces satisfying the condition. Next, as an application of the classification theorem, we give some examples of flat Riemannian metrics which are not conformal to the first fundamental form of any generic conformally flat hypersurface. These flat Riemannian metrics seem to provide counter-examples to Hertrich–Jeromin's claim [3, 5].