Splittings and Cr-structures for manifolds with nonpositive sectional curvature

Jianguo Cao, Jeff Cheeger, Xiaochun Rong

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Let M̃n denote the universal covering space of a compact Riemannian manifold, Mn, with sectional curvature, -1 ≤ KMn ≤ 0. We show that a collection of deck transformations of M̃n, satisfying certain (metric dependent) conditions, determines an open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ∈(n) > 0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov's "gap conjecture" in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.

Original languageEnglish (US)
Pages (from-to)139-167
Number of pages29
JournalInventiones Mathematicae
Volume144
Issue number1
DOIs
StatePublished - Jan 1 2001

Fingerprint

CR Structure
Nonpositive Curvature
Sectional Curvature
F-structure
Universal Space
Metric
Covering Space
Injectivity
Atlas
Isometric
Injective
Compact Manifold
Riemannian Manifold
Vanish
Radius
Denote
Subset
Dependent

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

@article{a86ad7f1edc0405aacc117a9c959da8c,
title = "Splittings and Cr-structures for manifolds with nonpositive sectional curvature",
abstract = "Let M̃n denote the universal covering space of a compact Riemannian manifold, Mn, with sectional curvature, -1 ≤ KMn ≤ 0. We show that a collection of deck transformations of M̃n, satisfying certain (metric dependent) conditions, determines an open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ∈(n) > 0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov's {"}gap conjecture{"} in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.",
author = "Jianguo Cao and Jeff Cheeger and Xiaochun Rong",
year = "2001",
month = "1",
day = "1",
doi = "https://doi.org/10.1007/PL00005800",
language = "English (US)",
volume = "144",
pages = "139--167",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",
number = "1",

}

Splittings and Cr-structures for manifolds with nonpositive sectional curvature. / Cao, Jianguo; Cheeger, Jeff; Rong, Xiaochun.

In: Inventiones Mathematicae, Vol. 144, No. 1, 01.01.2001, p. 139-167.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Splittings and Cr-structures for manifolds with nonpositive sectional curvature

AU - Cao, Jianguo

AU - Cheeger, Jeff

AU - Rong, Xiaochun

PY - 2001/1/1

Y1 - 2001/1/1

N2 - Let M̃n denote the universal covering space of a compact Riemannian manifold, Mn, with sectional curvature, -1 ≤ KMn ≤ 0. We show that a collection of deck transformations of M̃n, satisfying certain (metric dependent) conditions, determines an open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ∈(n) > 0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov's "gap conjecture" in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.

AB - Let M̃n denote the universal covering space of a compact Riemannian manifold, Mn, with sectional curvature, -1 ≤ KMn ≤ 0. We show that a collection of deck transformations of M̃n, satisfying certain (metric dependent) conditions, determines an open dense subset of Mn, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of Mn vanishes. We show that an abelian structure exists if the injectivity radius at all points of Mn is less than ∈(n) > 0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov's "gap conjecture" in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.

UR - http://www.scopus.com/inward/record.url?scp=0035627803&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035627803&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/PL00005800

DO - https://doi.org/10.1007/PL00005800

M3 - Article

VL - 144

SP - 139

EP - 167

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -