A metric characterization of Riemannian submersions
type: Item
instance of: scholarly article
title: A metric characterization of Riemannian submersions
zbMATH Open document ID: 0992.53025
DOI: 10.1023/A:1006683922481
author: Luis Guijarro
author: V. N. Berestovskij
published in: Annals of Global Analysis and Geometry
publication date: 2002-09-10T00:00:00Z
review text: According to the first author [Sib. Math. J. 28, No. 4, 552-562 (1987; Zbl 0643.53053); translation from Sib. Mat. Zh. 28, No. 4(164), 44-56 (1987)], a map between metric spaces is a submetry if it sends closed balls around a point to closed balls of the same radius, around the image point. In the paper under review, one proves that a submetry between smooth Riemannian manifolds is a \(C^{1,1}\) Riemannian submersion. The \(C^{1,1}\) regularity is optimal, since there is a submetry on the hyperbolic plane \(\mathbb{H}^{2}\rightarrow \mathbb{R}\) that is not \( C^{2}\). In the case of the metric projection of an open Riemannian manifold with nonnegative sectional curvature onto its soul, it is known that the projection is \(C^{2}\) and a.e. smooth [\textit{G. Perelman}, J. Differ. Geom. 40, 209-212 (1994; Zbl 0818.53056)]. The authors prove that if \(\varphi :M\rightarrow N\) is a submetry of complete Riemannian manifolds and \(M\) has a nonnegative sectional curvature, then \(N\) has also a nonnegative sectional curvature and if \(N\) is compact and \(M\) is flat, then \(N\) is flat. Another result of the paper uses the Cheeger-Gromoll splitting theorem [\textit{J. Cheeger} and \textit{D. Gromoll}, J. Differ. Geom. 6, 119-128 (1971; Zbl 0223.53033)] and states that if \(\varphi :M\rightarrow N\) is a submetry of complete Riemannian manifolds which have nonnegative Ricci curvatures, then there is a submetry \(\varphi _{0}:M_{0}\rightarrow N_{0}\) such that \(M\) and \(N\) are isometric with \(M_0\times \mathbb{R}^{n_{0}}\) and \(N_0\times \mathbb{R}^{n_{0}}\) respectively and \(\varphi \) can be identified with \( (\varphi _{0}\times Id)\). Defining the submetry with a soul as a submetry \(\varphi :M\rightarrow N\) for which there is a non-distance increasing map \(\psi :N\rightarrow M\) with \( \varphi \circ \psi =Id_{N}\), the authors prove some interesting results in the case where \(N\) is compact, extending obviously the case of the metric projection of an open …
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1563295
zbMATH Keywords: metric fibration
zbMATH Keywords: Riemannian submersion
zbMATH Keywords: submetry
MaRDI profile type: Publication
full work available at URL: a:1006683922481
OpenAlex ID: W1515640342
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description: scientific article; zbMATH DE number 1563295