an:04153387
Zbl 0703.53042
Fukaya, Kenji
Collapsing Riemannian manifolds to ones with lower dimension. II
EN
J. Math. Soc. Japan 41, No. 2, 333-356 (1989).
00168711
1989
j
53C23
sectional curvature; Hausdorff distance; infranilmanifold fibre; flat affine connections; minimal volume
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 \textit{M. Gromov}, Structure m??trique pour les vari??te?? riemannienne. Textes Math??matiques, 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 \textit{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 \(CN/CN\cap \Gamma \alpha \Aut \Gamma,\) 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.
Kenji Fukaya
Zbl 0478.53033; Zbl 0606.53027; Zbl 0509.53034; Zbl 0468.53036