## 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.## References

- E. Cartan,
*La déformation des hypersurfaces dans l’espace conforme réel à $n \ge 5$ dimensions*, Bull. Soc. Math. France**45**(1917), 57–121 (French). MR**1504762**, DOI 10.24033/bsmf.975 - Thomas E. Cecil and Patrick J. Ryan,
*Conformal geometry and the cyclides of Dupin*, Canadian J. Math.**32**(1980), no. 4, 767–782. MR**590644**, DOI 10.4153/CJM-1980-059-1 - Bang-yen Chen,
*Geometry of submanifolds*, Pure and Applied Mathematics, No. 22, Marcel Dekker, Inc., New York, 1973. MR**0353212** - N. H. Kuiper,
*On conformally-flat spaces in the large*, Ann. of Math. (2)**50**(1949), 916–924. MR**31310**, DOI 10.2307/1969587 - N. H. Kuiper,
*On compact conformally Euclidean spaces of dimension $>2$*, Ann. of Math. (2)**52**(1950), 478–490. MR**37575**, DOI 10.2307/1969480 - R. S. Kulkarni,
*Conformally flat manifolds*, Proc. Nat. Acad. Sci. U.S.A.**69**(1972), 2675–2676. MR**307113**, DOI 10.1073/pnas.69.9.2675 - Seiki Nishikawa and Yoshiaki Maeda,
*Conformally flat hypersurfaces in a conformally flat Riemannian manifold*, Tohoku Math. J. (2)**26**(1974), 159–168. MR**338967**, DOI 10.2748/tmj/1178241240 - Helmut Reckziegel,
*Completeness of curvature surfaces of an isometric immersion*, J. Differential Geometry**14**(1979), no. 1, 7–20 (1980). MR**577875** - Patrick J. Ryan,
*Homogeneity and some curvature conditions for hypersurfaces*, Tohoku Math. J. (2)**21**(1969), 363–388. MR**253243**, DOI 10.2748/tmj/1178242949 - Norman Steenrod,
*The Topology of Fibre Bundles*, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR**0039258**, DOI 10.1515/9781400883875

## 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