On nonpositively curved Euclidean submanifolds: splitting results
type: Item
instance of: scholarly article
title: On nonpositively curved Euclidean submanifolds: splitting results
zbMATH Open document ID: 0941.53014
author: Luis A. Florit
author: Fangyang Zheng
published in: Commentarii Mathematici Helvetici
publication date: 1999-05-25T00:00:00Z
review text: Let \(f\) be an isometric immersion of a Riemannian manifold \(M^{n}\) into Euclidean space \({\mathbb{R}}^{n+p}\). Denote by \(\nu(x)\) the dimension of the null space of the second fundamental form at \(x\). \(\nu\) is called the index of relative nullity of \(f\). It was shown by the first author of the paper [Math. Ann. 298, 187-192 (1994; Zbl 0810.53011)] that \(\nu \geq n-2p\) if \(M^{n}\) has nonpositive sectional curvature. In the present paper, the authors study immersions of minimal index. Namely, they prove that if \(M^{n}\) has nonpositive curvature and \(\nu=n-2p\), then, on an open dense subset of \(M^{n}\), \(f\) splits locally as a product of \(p\) nowhere flat Euclidean hypersurfaces of nonpositive sectional curvature. An analogous result for immersions into a (holomorphically) flat complex space form is also proved in the paper.
reviewed by: Mingliang Cai
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1278811
zbMATH Keywords: index of relative nullity
zbMATH Keywords: splitting
zbMATH Keywords: nonpositive sectional curvature
MaRDI profile type: Publication
full work available at URL: s000140050076
OpenAlex ID: W2095483824
DOI: 10.1007/S000140050076
description: scientific article; zbMATH DE number 1278811