Codimension three nonnegatively curved submanifolds with infinite fundamental group
type: Item
instance of: scholarly article
title: Codimension three nonnegatively curved submanifolds with infinite fundamental group
zbMATH Open document ID: 1228.53073
author: Juan-Miguel Gracia
published in: Mathematische Zeitschrift
publication date: 2011-03-02T00:00:00Z
review text: The authors investigate conditions on nonnegatively curved submanifolds of codimension three that imply the nonnegativity of the curvature operator. Their main result (Theorem 1.1) states that, if \(M^n \to {\mathbb{R}}^{n+3}\) is an isometric immersion of a complete Riemannian manifold \(M\) with nonnegative sectional curvature, then {\parindent6.5mm \begin{itemize}\item[(i)] if \(M\) has a line and \(f\) is noncylindrical (i.e., it does not split as an extrinsic product \(M_1\times {\mathbb{R}}^m\)), then \(M\) has nonnegative curvature operator; \item[(ii)] if \(M\) contains no lines and is covered by \(\widetilde M \times {\mathbb{R}}\), then the curvature operator of \(\widetilde M\) (and of \(M\)) is nonnegative. \end{itemize}} A result of \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. Math. (2) 96, 413--443 (1972; Zbl 0246.53049)] states that if the fundamental group of a compact manifold of nonnegative curvature is infinite then its universal covering splits isometrically as \(\widetilde M \times {\mathbb{R}}^k\), for \(k > 0\). Using this, the authors classify codimension-three isometric immersions of non-flat compact manifolds with nonnegative sectional curvatures and infinite fundamental group up to diffeomorphism.
reviewed by: Sergio Console
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 5859321
zbMATH Keywords: curvature operator
zbMATH Keywords: lines
zbMATH Keywords: nonnegative sectional curvature
zbMATH Keywords: fundamental group
MaRDI profile type: Publication
full work available at URL: s00209-009-0625-5
OpenAlex ID: W2088369752
Wikidata QID: Q126130606
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DOI: 10.1007/S00209-009-0625-5
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description: scientific article; zbMATH DE number 5859321