Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry

Jeff Cheeger, X. Rong

Research output: Contribution to journalArticle

Abstract

We study the class of n-Riemannian manifolds in the title such that the torsion elements in the fundamental group have a definite bound on their orders. Our main result asserts the existence of a kind of generalized Seifert fiber structure on Mn, for which the fundamental group of fibers injects into that of Mn. This provides a necessary and sufficient topological condition for a manifold to admit a sufficiently collapsed metric in our class. Among other consequences we obtain a strengthened version of the "gap conjecture" in this context.

Original languageEnglish (US)
Pages (from-to)141-163
Number of pages23
JournalGeometric and Functional Analysis
Volume5
Issue number2
DOIs
StatePublished - Mar 1995

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Fundamental Group
Riemannian Manifold
Covering
Fiber
Torsion
Sufficient
Metric
Necessary
Class
Context

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

Cite this

Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry. / Cheeger, Jeff; Rong, X.

In: Geometric and Functional Analysis, Vol. 5, No. 2, 03.1995, p. 141-163.

Research output: Contribution to journalArticle

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