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Transactions of the American Mathematical Society

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


References [Enhancements On Off] (What's this?)

  • [1] E. Cartan, La déformation des hypersurfaces dans l'espace conforme réel a $ n \geqslant 5$ dimensions, Bull. Soc. Math. France 45 (1917), 57-121. MR 1504762
  • [2] T. Cecil and P. Ryan, Conforma geometry and the cyclides of Dupin, Canad. J. Math. 32 (1980), 767-782. MR 590644 (82f:53004)
  • [3] B. Y. Chen, Geometry of submanifolds, M. Dekker, New York, 1973. MR 0353212 (50:5697)
  • [4] N. Kuiper, On conformally flat spaces in the large, Ann. of Math. (2) 50 (1949), 916-924. MR 0031310 (11:133b)
  • [5] -, On compact conformally euclidean spaces of dimension $ > 2$, Ann. of Math. (2) 52 (1950), 478-490. MR 0037575 (12:283c)
  • [6] R. S. Kulkarni, Conformally flat manifolds, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2675-2676. MR 0307113 (46:6234)
  • [7] S. Nishikawa and Y. Maeda, Conformally flat hypersurfaces in a conformally flat Riemannian manifold, Tôhoku Math. J. 26 (1974), 159-168. MR 0338967 (49:3730)
  • [8] H. Reckziegel, Completeness of curvatures surfaces of an isometric immersion, J. Differential Geom. 14 (1979), 7-20. MR 577875 (81f:53045)
  • [9] P. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tôhoku Math. J. 21 (1969), 363-388. MR 0253243 (40:6458)
  • [10] N. Steenrod, Topology of fiber bundles, Princeton Univ. Press, Princeton, N.J., 1951. MR 0039258 (12:522b)

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DOI: https://doi.org/10.1090/S0002-9947-1985-0773056-0
Article copyright: © Copyright 1985 American Mathematical Society

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