The hyperbolic n-space as a graph in Euclidean (6n-6)-space
type: Item
instance of: scholarly article
title: The hyperbolic n-space as a graph in Euclidean (6n-6)-space
zbMATH Open document ID: 0612.53034
DOI: 10.1007/BF01171261
author: Wolfgang Henke
author: Wolfgang Nettekoven
published in: Manuscripta Mathematica
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155236
review text: Let \(H^ n\) denote the n-dimensional hyperbolic space of constant curvature (-1) and \(E^ N\) the N-dimensional Euclidean space. \textit{D. Blanusa} [Monatsh. Math. 59, 217-229 (1955; Zbl 0067.144)] constructed an isometric \(C^{\infty}\)-imbedding \(H^ 2\to E^ 6\) whose image is the graph of a \(C^{\infty}\)-map \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 4\). For \(n\geq 3\), in the same article, Blanusa was only able to construct a 1-1 isometric \(C^{\infty}\)-immersion \(H^ n\to E^{6n-5}\) which is not an imbedding in the strong sense (i.e. not a homeomorphism onto a topological subspace). The present paper generalizes the stronger 2- dimensional result of Blanusa. Theorem: For each \(n\geq 2\), there exists an isometric \(C^{\infty}\)-imbedding \(H^ n\to E^{6n-6}\) whose image is the graph of a C-map \({\mathbb{R}}^ n\to {\mathbb{R}}^{5n-6}\). Moreover explicit formulas are given which apply to the isometric imbedding problem for a larger class of Riemannian manifolds.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 3990294
zbMATH Keywords: hyperbolic space
zbMATH Keywords: Euclidean space
zbMATH Keywords: graph
zbMATH Keywords: isometric imbedding
MaRDI profile type: Publication
cites work: Über die Einbettung hyperbolischer Räume in euklidische Räume
cites work: Isometrische Immersionen des \(n\)-dim. hyperbolischen Raumes \(H^n\) in \(E^{4n-3}\)
OpenAlex ID: W1972671665
description: scientific article; zbMATH DE number 3990294