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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $ {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.

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Article copyright: © Copyright 1985 American Mathematical Society