Manfredo P. do Carmo – Selected Papers pp 237-251 | Cite as

# 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.

## Keywords

Riemannian Manifold Local Orientation Betti Number Conformal Space Flat Space## Preview

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