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Collapsing Riemannian manifolds to ones with lower dimension. II. (English) Zbl 0703.53042
This paper is a continuation of the author’s paper with the same title [Part I, J. Differ. Geom. 25, 139–156 (1987; Zbl 0606.53027)]. We discuss the topological properties of Riemannian manifolds \(M\) the absolute value of their sectional curvature is smaller than \(1\) and which \(M\) are close to a Riemannian manifold \(X\) (of lower dimension) with respect to the Hausdorff distance [see M. Gromov, Structure métrique pour les variéteś riemannienne. Textes Mathématiques, 1. Paris: Cedic/Fernand Nathan (1981; Zbl 0509.53034)]. In the former paper, it has been proved that \(M\) fibres over \(X\) with an infranilmanifold fibre, \(N/\Gamma\).
In the present paper, employing E. A. Ruh’s technique [J. Differ. Geom. 17, 1–14 (1982; Zbl 0468.53036)] a smooth family of flat affine connections on the fibres is constructed. As a consequence, the structure group of the fibration is reduced to the semi-direct product \(CN/CN\cap \Gamma \alpha \operatorname{Aut} \Gamma,\) where \(CN\) is the center of the nilpotent group \(N\). It turns out that this condition on the structure group is sufficient to construct a family of metrics on \(M\) converging to \(X\). An application to a gap phenomenon of minimal volume (diameter bound) of aspherical manifolds is given.
Reviewer: Kenji Fukaya

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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