On embedded flat surfaces in \(S^3\)
type: Item
instance of: scholarly article
title: On embedded flat surfaces in \(S^3\)
zbMATH Open document ID: 0917.53023
DOI: 10.1007/BF02921704
author: Jiri Dadok
author: Ji-Ping Sha
published in: The Journal of Geometric Analysis
publication date: 1999-08-01T00:00:00Z
review text: There are many results on immersed flat surfaces \(M\) in the standard three-dimensional sphere \(S^3\) of constant sectional curvature one, some of which are reviewed (with slightly simplified proofs) in the first section of the present paper. However, the question of which of these surfaces are embedded was not addressed in these earlier results. This is the main concern of the authors in the present paper and they prove, among other results, that every embedded flat torus \(M \subset S^3\) has an antipodal symmetry and double covers an embedded torus in \({\mathbb{R}}P^3\). They also show how to construct every embedded flat torus \(M \subset S^3\), or \({\mathbb{R}}P^3\), from a pair of curves in the 2-sphere \(S^2\) satisfying a suitable condition. Finally, they show that there are embedded flat cylinders in \(S^3\), but they conjecture that it is impossible to embed \({\mathbb{R}}^2\) isometrically into \(S^3\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 1192503
zbMATH Keywords: flat surfaces in the three sphere
zbMATH Keywords: constant sectional curvature
zbMATH Keywords: imbedded surfaces
zbMATH Keywords: imbeddability
reviewed by: Vagn Lundsgaard Hansen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1192503