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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Compact conformally flat hypersurfaces
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by Manfredo do Carmo, Marcos Dajczer and Francesco Mercuri PDF
Trans. Amer. Math. Soc. 288 (1985), 189-203 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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