A correspondence for isometric immersions into product spaces and its applications
type: Item
instance of: scholarly article
title: A correspondence for isometric immersions into product spaces and its applications
zbMATH Open document ID: 1202.53059
author: José A. Gálvez
author: Asun Jiménez
author: Pablo Mira
published in: Journal of Geometry and Physics
publication date: 2010-09-24T00:00:00Z
review text: Let \(\psi : M^n \to N^n \times P^r\) be an isometric immersion from an \(n\)-dimensional semi-Riemannian manifold \(M^n\) into a product manifold \(N^n \times P^r\) equipped with a semi-Riemannian warped product metric \(g_N \otimes_{\mathcal F} g_P\). Assume that \(\psi\) is a graph, that is, \(\psi\) is of the form \(\psi = (\pi,h \circ \pi)\), where \(\pi\) is a diffeomorphism from \(M^n\) into some open domain \(\Omega \subset N^n\), and \(h : \Omega \to P^r\) is smooth. Then \(\widehat{\psi} = (\pi^{-1},h)\) is an isometric immersion from \(\Omega \subset N^n\) into the product manifold \(M^n \times P^r\) equipped with the semi-Riemannian metric \(g^\prime = \frac{1}{{\mathcal F}^2}(g_M \otimes (-g_P))\). The authors investigate the correspondence between the two maps \(\psi\) and \(\widehat{\psi}\). As a consequence they obtain several existence and non-existence results for hypersurfaces in Riemannian or Lorentzian spaces. For example, if \(M^n\) is a complete Riemannian manifold with negative Ricci curvature whose scalar curvature is bounded from above by a negative constant, they conclude that the Euclidean space \({\mathbb R}^n\) cannot be isometrically immersed into the Lorentzian product space \(M^n \times {\mathbb L}\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 5791217
zbMATH Keywords: warped product
zbMATH Keywords: isometric immersion
zbMATH Keywords: graph
Wikidata QID: Q56885368
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
full work available at URL: j.geomphys.2010.07.001
OpenAlex ID: W2090348009
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DOI: 10.1016/J.GEOMPHYS.2010.07.001
description: scientific article; zbMATH DE number 5791217