Compact Conformally Flat Hypersurfaces

Manfredo do Carmo, Marcos Dajczer and Francesco Mercuri
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Manfredo do Carmo: IMPA
Marcos Dajczer: IMPA
Francesco Mercuri: Instituto de Matemática, UNICAMP

A chapter in Manfredo P. do Carmo – Selected Papers, 2012, pp 237-251 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-25588-5_19

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DOI: 10.1007/978-3-642-25588-5_19

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