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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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