Explicit representation of compact conformally flat hypersurfaces
Diffeomorphism types of compact conformally flat hypersurfaces of dimension greater or equal than 4, in a simply connected space form, were studied by \textit{M. doCarmo, M. Dajczer} and \textit{F. Mercuri} [Trans. Am. Math. Soc. 288, 189-203 (1985; Zbl 0554.53040)]. Moreover, their intrinsic conformal geometry was studied by \textit{U. Pinkall} [Aspects Math., E 12, 217-236 (1988; Zbl 0657.53030)], establishing that they are conformally equivalent either to the sphere or to a classical Schottky manifold. Assuming that the ambient space is the \(n+1\)-dimensional sphere, the author constructs explicit examples of compact conformally flat hypersurfaces for \(n\) greater than three. He determines their conformal type by showing the explicit conformal diffeomorphisms between them and the classical Schottky manifolds. He also gives examples of conformally flat tori in \(S^{n+1}\) with umbilic points, and conformally flat manifolds diffeomorphic to the Klein bottle, with umbilic points.
scientific article; zbMATH DE number 1230530