Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature
type: Item
instance of: scholarly article
title: Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature
zbMATH Open document ID: 0682.53055
author: Toru Ishihara
published in: Michigan Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: The maximal spacelike submanifolds of a pseudo-Riemannian manifold of constant curvature are investigated. Let \(N_ p^{n+p}(c)\) denote an \((n+p)\)-dimensional pseudo-Riemannian manifold of constant curvature \(c\) and index \(p\). Let \(M\) be an \(n\)-dimensional complete Riemannian manifold which is isometrically immersed in \(N_ p^{n+p}(c)\) and maximal, i.e. has vanishing mean curvature. Let \(S\) be the square of the length of the second fundamental form. Three main results are proved. \par (i) If \(c\ge 0\) then the immersion is totally geodesic \((S=0)\) and \(M\) has constant curvature \(c\). \par (ii) If \(c<0\) then \(0\le S\le -npc.\) In the case \(c=-1/r^ 2,\) the ambient space \(N_p^{n+p}(c)\) is a pseudo-hyperbolic space \(H_p^{n+p}(r).\) \par (iii) The only complete connected maximal spacelike submanifolds of dimension \(n\) in \(H_p^{n+p}(1)\) satisfying \(S=np\) are the submanifolds \(H^{n_1}(\sqrt{n_1/n})\times\ldots\times H^{n_{p+1}}(\sqrt{n_{p+1}/n}),\) which can be described as follows. Let \(H^ m(r)\) denote the component of \(H^m_0(r)\) through \((0,0,\ldots,0,r)\). Let \(n_1,\ldots,n_{p+1}\) be positive integers with \(n=n_1+\ldots+n_{p+1}.\) For \(x_ i\in H^{n/i}(\sqrt{n_i/n}),\) \(x=(x_1,\ldots,x_{p+1})\) is a vector in \({\mathbb{R}}_{p+1}^{n+p+1}\) with \(<x,x>=-1.\) This defines an isometric immersion of \par \[ H^{n_1}(\sqrt{n_ 1/n})\times\ldots\times H^{n_{p+1}}(\sqrt{n_{p+1}/n})\) \text{ into } \(H_p^{n+p}(1). \] These results can be compared with the work of \textit{S. S. Chern}, \textit{M. P. do Carmo}, and \textit{S. Kobayashi} [Functional analysis and related fields, Conf. Chicago 1968, 59--75 (1970; Zbl 0216.44001)] on minimal submanifolds of Riemannian spheres. Result (i) above generalizes the theorem of \textit{S.-Y. Cheng} and \textit{S.-T. Yau} [Ann. Math. (2) 104, 407--419 (1976; Zbl 0352.53021)] that complete maximal spacelike hypersurfaces in Minkowski space are totally geodesic.
reviewed by: Dean E. Allison
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 4117401
zbMATH Keywords: maximal spacelike submanifolds
zbMATH Keywords: pseudo-Riemannian manifold
zbMATH Keywords: constant curvature
zbMATH Keywords: vanishing mean curvature
zbMATH Keywords: totally geodesic
Wikidata QID: Q115238999
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003815
OpenAlex ID: W2015883595
DOI: 10.1307/MMJ/1029003815
description: scientific article; zbMATH DE number 4117401