Submetries of space forms of nonnegative curvature
type: Item
instance of: scholarly article
title: Submetries of space forms of nonnegative curvature
zbMATH Open document ID: 0643.53053
DOI: 10.1007/BF00973842
published in: Siberian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The author introduces the following notion of a submetry: a map \(p: M\to N\) of metric spaces is called a submetry if for every point \(x\in M\) the image of each closed ball with the center at x under the map p is the closed ball of the same radius with the center at the point p(x). Every Riemannian submersion of a complete Riemannian manifold is a submetry. The following main results are established. Theorem 1. Any submetry \(p: M\to N\) of complete Euclidean spaces M and N (possibly infinite dimensional) can be represented as a composition \(p=i_ 1\circ p_ 1\), where \(p_ 1\) is an orthogonal projection of M on a closed Euclidean subspace \(M_ 1\subset M\) and \(i_ 1: M_ 1\to N\) is an isometry. Theorem 2. Any submetry \(p: M\to N\) of unit spheres of complete Euclidean spaces is an isometry. Furthermore a connection between special Riemannian submersions and the existence of a transversal foliation of a given bundle is established in the paper. The author proves that the Hopf bundles do not have transversal foliations.
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4048589
zbMATH Keywords: nonnegative curvature
zbMATH Keywords: map of metric spaces
zbMATH Keywords: submetry of complete Euclidean spaces
zbMATH Keywords: Riemannian submersion
zbMATH Keywords: isometry
zbMATH Keywords: unit spheres of complete Euclidean spaces
zbMATH Keywords: transversal foliation
author: V. N. Berestovskij
reviewed by: Sergey A. Antonyan
MaRDI profile type: Publication
cites work: The fundamental equations of a submersion
cites work: A Sufficient Condition that a Mapping of Riemannian Manifolds be a Fibre Bundle
cites work: Q3706950
cites work: Beginner's course in topology. Geometric chapters. Transl. from the Russian by A. Iacob
description: scientific article; zbMATH DE number 4048589