Explicit representation of compact conformally flat hypersurfaces
type: Item
instance of: scholarly article
title: Explicit representation of compact conformally flat hypersurfaces
zbMATH Open document ID: 0951.53045
author: Yoshihiko Suyama
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-01-27T00:00:00Z
review text: Diffeomorphism types of compact conformally flat hypersurfaces of dimension greater or equal than 4, in a simply connected space form, were studied by \textit{M. doCarmo, M. Dajczer} and \textit{F. Mercuri} [Trans. Am. Math. Soc. 288, 189-203 (1985; Zbl 0554.53040)]. Moreover, their intrinsic conformal geometry was studied by \textit{U. Pinkall} [Aspects Math., E 12, 217-236 (1988; Zbl 0657.53030)], establishing that they are conformally equivalent either to the sphere or to a classical Schottky manifold. Assuming that the ambient space is the \(n+1\)-dimensional sphere, the author constructs explicit examples of compact conformally flat hypersurfaces for \(n\) greater than three. He determines their conformal type by showing the explicit conformal diffeomorphisms between them and the classical Schottky manifolds. He also gives examples of conformally flat tori in \(S^{n+1}\) with umbilic points, and conformally flat manifolds diffeomorphic to the Klein bottle, with umbilic points.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 1230530
zbMATH Keywords: conformally flat submanifold
zbMATH Keywords: hypersurface
zbMATH Keywords: real space form
reviewed by: Oscar J. Garay
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224972
OpenAlex ID: W2160530529
cites work: Compact Conformally Flat Hypersurfaces
cites work: Q3808798
cites work: Q3810460
cites work: Q3805409
cites work: Completeness of curvature surfaces of an isometric immersion
DOI: 10.2748/TMJ/1178224972
description: scientific article; zbMATH DE number 1230530