Complex hyperbolic manifolds and exotic smooth structures
type: Item
instance of: scholarly article
title: Complex hyperbolic manifolds and exotic smooth structures
zbMATH Open document ID: 0804.53055
DOI: 10.1007/BF01232234
published in: Inventiones Mathematicae
publication date: 1994-07-04T00:00:00Z
full work available at URL: https://eudml.org/doc/144208
review text: In earlier work the authors considered the problem of changing the smooth structure on a real hyperbolic manifold while equipping the new structure with a negatively curved Riemannian metric. One technique which was employed was the use of an exotic homotopy sphere flow \(\theta_{n-1}\). In this paper, the problem for compact hyperbolic manifolds is considered. In particular, the authors show that for any positive integer \(m\) of the form \(4n+1\), \(n\in\mathbb{Z}\) (or \(m=4\)) there exist a pair of compact smooth manifolds \(M^{2m}\) and \(N^{2m}\) such that (a) \(M^{2m}\) and \(N^{2m}\) are homeomorphic but not diffeomorphic, (b) \(M^{2m}\) is a complex hyperbolic manifold of complex dimension \(m\), and (c) \(N^{2m}\) supports a negatively curved Riemannian metric. Moreover, given \(\varepsilon>0\), the manifolds can be constructed so that all of the sectional curvatures of \(N^{2m}\) lie in a closed interval \([-4- \varepsilon,-1+\varepsilon]\).
reviewed by: A. P. Stone
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57R55
zbMATH DE Number: 599646
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: sectional curvatures
author: F. Thomas Farrell
author: Lowell Jones
MaRDI profile type: Publication
cites work: On the groups \(J(X)\). IV
cites work: Q5518126
cites work: Compact Clifford-Klein forms of symmetric spaces
cites work: Curvature operators: pinching estimates and geometric examples
cites work: Homotopy equivalences of almost smooth manifolds
cites work: Q4067705
cites work: Q4078829
cites work: Negatively Curved Manifolds with Exotic Smooth Structures
cites work: Q3138467
cites work: Nonuniform hyperbolic lattices and exotic smooth structures
cites work: Kähler manifolds and 1/4-pinching
cites work: Some applications of the Weil representation
cites work: Groups of homotopy spheres. I
cites work: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. (AM-88)
cites work: Residually finite groups
cites work: Characteristic Classes. (AM-76)
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Automorphic forms and the periods of abelian varieties
cites work: Negatively \(1\over 4\)-pinched Riemannian metric on a compact Kähler manifold
OpenAlex ID: W2016959124
Recommended article: Almost complex manifolds and hyperbolicity
Recommended article: Hypercomplex manifolds and hyperholomorphic bundles
Recommended article: Almost complex structures on hyperbolic manifolds
Recommended article: Fueter and hypercomplex structures on smooth manifolds
Recommended article: Q5703406
Recommended article: Hyperbolic manifolds
Recommended article: Q4089110
Recommended article: Q3378424
Recommended article: Geometry and topology of complex hyperbolic and Cauchy-Riemannian manifolds
Recommended article: Q3056428
description: scientific article; zbMATH DE number 599646