Compact conformally flat hypersurfaces

Abstract

Roughly speaking, a conformal space is a differentiable manifold in which the notion of angle of tangent vectors at a point makes sense and varies differentiably with p; two such spaces are (locally) equivalent if they are rela ted by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space Rn. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure: in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to givea description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, asume the ambient space to be R n+1. Then, if a conformally flat hypersurface 1 can be described as follows. Diffeomorphically, M nis a sphere S nwith h1(M) handles attached, where h1 (M) is the first Betti number of M. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of Rn+1that are foliated by complete round (n-1)-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of Rn+1: (a) an open piece of an n-sphere or an n-plane bounded by round (n-1)-sphere, (b) a round (n-1)-sphere, (c) a point.