Compact Conformally Flat Hypersurfaces

  • Manfredo do Carmo
  • Marcos Dajczer
  • Francesco Mercuri


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.


Riemannian Manifold Local Orientation Betti Number Conformal Space Flat Space 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manfredo do Carmo
    • 1
  • Marcos Dajczer
    • 1
  • Francesco Mercuri
    • 2
  1. 1.IMPARio de JaneiroBrasil
  2. 2.Instituto de Matemática, UNICAMPSãO PauloBrasil

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