Conformal hypersurfaces with the same Gauss map

Abstract

In this paper we provide a complete classification of all hypersurfaces of Euclidean space which admit conformal deformations, other than the ones obtained through conformal diffeomorphisms of the ambient space, preserving the Gauss map. Elie Cartan, in one of his earlier papers in differential geometry, classified all Euclidean hypersurfaces, of dimension at least five, which admit conformal deformations other than the trivial ones obtained through compositions with conformal diffeomorphisms of the ambient space. When all possible conformal deformations are trivial, a hypersurface is called conformally rigid. First, Cartan proved that a hypersurface is conformally rigid if at no point does there exist a principal curvature of multiplicity at least n 2 (see [Ca], [CD2], or [Da]). Using this, he concluded that a conformally deformable hypersurface is either conformally flat or a 2-parameter envelope of spheres or planes of some very special type. Moreover, the set of all conformal deformations is either a 1parameter family or there is just one other deformation. In this paper we classify all Euclidean hypersurfaces which admit nontrivial conformal deformations preserving the Gauss map. In the isometric case, a complete classification for any codimension has been obtained by Dajczer and Gromoll [DG2], while the conformal case, but only for surfaces, is due to Vergasta [Ve2]. Theorem. Let f, g: Mn -+ Rn+1, n > 3, be conformal immersions of an ndimensional connected Riemannian manifold with the same Gauss map. Assume that on no open subset do they differ by a conformal diffeomorphism of Rn+l. Then f(Mn) is part of one of the following examples while g(Mn) is of the same type: (i) a minimal real Kaehler hypersurface, (ii) a rotation hypersurface over a plane curve, (iii) a rotation hypersurface over a minimal surface in R3. Minimal real Kaehler hypersurfaces have been completely classified by Dajczer and Gromoll in [DG2]. The assumption that f and g do not differ locally by a conformal deformation of the ambient space has been introduced in order to produce a global result. Without that assumption, our proof shows that the only other possibility is to have an open subset where, up to homothety Received by the editors December 10, 1993. 1991 Mathematics Subject Classification. Primary 53C42. ? 1995 American Mathematical Society