### Abstract

Let M̃^{n} denote the universal covering space of a compact Riemannian manifold, M^{n}, with sectional curvature, -1 ≤ K_{Mn} ≤ 0. We show that a collection of deck transformations of M̃^{n}, satisfying certain (metric dependent) conditions, determines an open dense subset of M^{n}, 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 M^{n} vanishes. We show that an abelian structure exists if the injectivity radius at all points of M^{n} 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 language | English (US) |
---|---|

Pages (from-to) | 139-167 |

Number of pages | 29 |

Journal | Inventiones Mathematicae |

Volume | 144 |

Issue number | 1 |

DOIs | |

State | Published - 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Inventiones Mathematicae*,

*144*(1), 139-167. https://doi.org/10.1007/s002220000120

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

Research output: Contribution to journal › Article

*Inventiones Mathematicae*, vol. 144, no. 1, pp. 139-167. https://doi.org/10.1007/s002220000120

}

TY - JOUR

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

AU - Cao, Jianguo

AU - Cheeger, Jeff

AU - Rong, Xiaochun

PY - 2001

Y1 - 2001

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 - 10.1007/s002220000120

DO - 10.1007/s002220000120

M3 - Article

AN - SCOPUS:0035627803

VL - 144

SP - 139

EP - 167

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -