Isometric immersions into \(\mathbb{S}^m \times \mathbb{R}\) and \(\mathbb{H}^m \times \mathbb{R}\) with high codimensions
type: Item
instance of: scholarly article
title: Isometric immersions into \(\mathbb{S}^m \times \mathbb{R}\) and \(\mathbb{H}^m \times \mathbb{R}\) with high codimensions
zbMATH Open document ID: 1193.53134
author: Qun Chen
author: Qing Cui
published in: Results in Mathematics
publication date: 2010-06-23T00:00:00Z
review text: \textit{B. Daniel} [Trans. Am. Math. Soc. 361, No.~12, 6255--6282 (2009; Zbl 1213.53075)], gave a necessary and sufficient condition for an \(n\)-dimensional Riemannian manifold to be isometrically immersed in \(S^n \times {\mathbb R}\) or \(H^n \times {\mathbb R}\) in terms of its first and second fundamental forms and some other conditions. In the present paper the authors generalise this to higher codimensions. The necessary and sufficient conditions are the three equations of Gauss, Codazzi and Ricci, together with two conditions on the vector field on \(M\) which is obtained by projection of the vector field \(\partial/\partial t\) associated to the \({\mathbb R}\)-factor in the target space.
reviewed by: Jürgen Berndt
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 5725192
zbMATH Keywords: isometric immersions
zbMATH Keywords: fundamental equations of submanifolds
MaRDI profile type: Publication
full work available at URL: s00025-010-0031-7
OpenAlex ID: W2044371068
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DOI: 10.1007/S00025-010-0031-7
description: scientific article; zbMATH DE number 5725192