Isometric immersions of pseudo-Riemannian space forms
type: Item
instance of: scholarly article
title: Isometric immersions of pseudo-Riemannian space forms
zbMATH Open document ID: 1077.53051
author: Chen Qing
author: Dafeng Zuo
author: Cheng Yi
published in: Journal of Geometry and Physics
publication date: 2005-06-13T00:00:00Z
review text: Let \(f: M^n_s(K)\to N^{2n-1}_{s+q}(c)\) be a local isometric immersion of a time-like submanifold \(M^n_s(K)\) with constant sectional curvature \(K\) and index \(s\) into a pseudo-Riemannian space form \(N^{2n-1}_{s+q}(c)\) with constant sectional curvature \(c\) and index \(s+ q\), where \(q\geq 0\), \(1\leq s\leq n-1\) and \(K\neq c\). These immersions are studied by means of the moving frames adapted to \(M^n_s(K)\) in \(N^{2n-1}_{s+q}(c)\) using Cartan's method, structural equations etc. A theorem by \textit{J. L. Barbosa}, \textit{W. Ferreira} and \textit{K. Tenenblat} [Ann. Global Anal. Geom. 14, No. 4, 381--401 (1996; Zbl 0866.53049)] is generalized to this case. Namely, assume that the second fundamental form is orthogonally diagonalizable and there exists a point \(p\) of \(M\) where the principal normal curvatures are different from \(K- c\). Then it is established that on an open contractible region \(U\) of \(p\), there exist line of curvature coordinates \(\{x_1,\dots, x_n\}\) and local parallel normal frame fields, for which the first and second fundamental forms have the diagonal forms. Also, the corresponding Gauss-Codazzi-Ricci equations are derived. The same is done for the particular cases of \(K< c\) and \(q= 0\), and of \(K> c\) and \(q= 0\), where the normal bundle is flat. The higher-dimensional generalizations of the classical Bäcklund theorem are given. By introducing the Chebyshev coordinates and using the Bäcklund theorem an explicit Bäcklund transformation is given, as well as a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58J72
Mathematics Subject Classification ID: 37K35
Mathematics Subject Classification ID: 35A30
zbMATH DE Number: 2175254
zbMATH Keywords: isometric immersion
zbMATH Keywords: pseudo-Riemannian space forms
zbMATH Keywords: Bäcklund theorem
zbMATH Keywords: Bäcklund transformation
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
full work available at URL: j.geomphys.2004.03.002
OpenAlex ID: W2034469333
Wikidata QID: Q115353527
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DOI: 10.1016/J.GEOMPHYS.2004.03.002
description: scientific article; zbMATH DE number 2175254