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 related 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, assume the ambient space to be ${R^{n + 1}}$. Then, if $n \geqslant 4$, a conformally flat hypersurface ${M^n} \subset {R^{n + 1}}$ can be described as follows. Diffeomorphically, ${M^n}$ is a sphere ${S^n}$ with ${b_1}(M)$ handles attached, where ${b_1}(M)$ 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.