The holonomy in open manifolds of nonnegative curvature
type: Item
instance of: scholarly article
title: The holonomy in open manifolds of nonnegative curvature
zbMATH Open document ID: 0886.53030
published in: Michigan Mathematical Journal
publication date: 1998-03-18T00:00:00Z
review text: Let \(V^n\) be a complete noncompact manifold of nonnegative sectional curvature \(K_\sigma\geq 0\). Let \(S\) be a soul of \(V^n\), i.e., \(S\) is a compact, totally convex, totally geodesic submanifold of \(V^n\), without boundary, such that \(0\leq\dim S\leq\dim V^n\). \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. Math., II. Ser. 96, 413-443 (1972; Zbl 0246.53049)] proved that \(V^n\) is diffeomorphic to the normal bundle \(\nu(S)\) and, if V\(^n\) is isometric to the direct product \(V^n=S\times W\) (where \(W\) is a complete noncompact manifold of nonnegative sectional curvature, diffeomorphic to the Euclidean space), then the holonomy operator is the identity, i.e., \(\nu(S)\) has trivial holonomy. The principal theorem in this article states that the converse of this last statement is also true, i.e., if \(\nu(S)\) has trivial holonomy, then \(V^n\) is isometric to the direct product \(V^n=S\times W\) where \(W\) is a complete noncompact manifold, diffeomorphic to the Euclidean space of the corresponding dimension. According to the author, this theorem has already been stated [\textit{V. B. Marenich}, Sov. Math. Dokl. 24, 595-597 (1981; Zbl 0495.53037)]. In the present article a clear and detailed proof is given. The second part of this work includes new proofs of three theorems. The original ones were given by the author [in ``The holonomy in open manifolds of nonnegative curvature'', MSRI preprint No. 003-94], but were very long. G. Perel'man gave a new proof for one of these statements, more precisely, for the theorem that states that for every point \(p\in S\) and every 2-dimensional direction \(\sigma\) that is normal to \(S\) at this point (i.e., \(\sigma\subset\nu_p(S)\)), then \(I_\omega= \text{id}\) for every contractible curve \(\omega\) on \(S\), if \(K_\sigma=0\), and the universal cover \(\widetilde V^n\) of \(V^n\) is isometric to the direct product (\(I_\omega\) is the parallel translation along this curve). The author includes the proof gi…
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 934197
zbMATH Keywords: complete noncompact manifold
zbMATH Keywords: nonnegative sectional curvature
zbMATH Keywords: holonomy
zbMATH Keywords: soul
author: Valery Borisovich Marenich
reviewed by: Ana Pereira do Vale
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005462
OpenAlex ID: W2160357752
DOI: 10.1307/MMJ/1029005462
description: scientific article; zbMATH DE number 934197