Compact conformally flat hypersurfaces

Authors:
Manfredo do Carmo, Marcos Dajczer and Francesco Mercuri

Journal:
Trans. Amer. Math. Soc. **288** (1985), 189-203

MSC:
Primary 53C40

MathSciNet review:
773056

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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 ; 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 . 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 . Then, if , a conformally flat hypersurface can be described as follows. Diffeomorphically, is a sphere with handles attached, where is the first Betti number of . Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of that are foliated by complete round -spheres and are joined through their boundaries to the following three types of umbilic submanifolds of : (a) an open piece of an -sphere or an -plane bounded by round -sphere, (b) a round -sphere, (c) a point.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0773056-0

Article copyright:
© Copyright 1985
American Mathematical Society