Compact Conformally Flat Hypersurfaces

Abstract

Roughly speaking, a conformal space is a differentiable manifold $M^n$in which the notion of angle of tangent vectors at a point $p \in M^n$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 R ⁿ. 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 give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, as~ume the ambient space to be R ⁿ⁺¹. Then, if $n \geqslant 4$, a conformally flat hypersurface ${M}^{n} \subset {R}^{n+1}$ 1 can be described as follows. Diffeomorphically, M ⁿ is a sphere S ⁿ with 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 R ⁿ⁺¹ that are foliated by complete round (n – 1 )-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of R ⁿ⁺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.

1980 Mathematics Subject Classification. Primary 53A30: Secondary 53C40.