A splitting theorem for nonnegatively curved Alexandrov spaces
type: Item
instance of: scholarly article
title: A splitting theorem for nonnegatively curved Alexandrov spaces
zbMATH Open document ID: 1261.53044
author: Andreas Wörner
published in: Geometry \& Topology
publication date: 2013-02-25T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional compact Alexandrov space of nonnegative curvature. A ``boundary stratum'' of \(M\) is an extremal set of locally constant codimension \(1\). A ``stratification'' of \(\partial M\) is a cover of \(\partial M\) by boundary strata whose pairwise intersections have codimension at least \(2\). The main result of the paper (Theorem 1.2) infers the metric product structure of \(M\) from the existence of a particular stratification of \(\partial M\). Namely, assume that \(F_1,\dots, F_{\ell+1}\) is a stratification for which there exists \(k\in\{1,\dots,\ell\}\) such that (i) the intersection \(F_1\cap\dots\cap F_{k+1}\) is empty; (ii) if any of the sets \(F_1,\dots,F_{k+1}\) is omitted from (i), the resulting intersection is nonempty. Then the conclusion is that \(M\) is isometric to the product \(S\times D\) where both factors are nonnegatively curved Alexandrov spaces, \(\dim S=n-k\), and \(\dim D=k\). Here, \(S\) can be chosen to be any of the intersections mentioned in (ii) above. Moreover, in the special case \(k=\ell\) the set \(S\) is a soul of \(M\). Among several other corollaries, the author obtains the following: If \(M\) has at least \(n+2\) boundary strata, then it is a nontrivial metric product. The proofs employ a diverse array of methods of metric geometry and are carefully presented with precise pointers to the literature. Additional details are available in the author's thesis on which this article is based.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 51H25
zbMATH DE Number: 6138583
zbMATH Keywords: Alexandrov space
zbMATH Keywords: nonnegative curvature
zbMATH Keywords: metric splitting
zbMATH Keywords: extremal subset
zbMATH Keywords: soul
zbMATH Keywords: boundary strata
reviewed by: Leonid V. Kovalev
MaRDI profile type: Publication
full work available at URL: gt.2012.16.2391
OpenAlex ID: W1993647933
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DOI: 10.2140/GT.2012.16.2391
description: scientific article; zbMATH DE number 6138583