### Abstract

Let M^{n} denote a closed Riemannian manifold with nonpositive sectional curvature. Let X^{n} denote a closed smooth manifold which admits an F-structure, F. If there exists f : X^{n} → M ^{n} with nonzero degree, then M^{n} has a local splitting structure S: 1) The universal covering space with the pull-back metric, has a locally finite covering by closed convex subsets, each of which splits isometrically as a product with nontrivial Euclidean factor. 2) This collection of sets and splittings are invariant under the group of covering transformations. 3) The projection to M^{n} of any flat (i.e. Euclidean slice) of S is a closed immersed submanifold. The structures, F, S, satisfy a consistency condition. If F is injective, all orbits have dimension ≥ n - 2 and f induces an isomorphism of fundamental groups, then S is abelian i.e. for all p ∈ M^{n}, there is a flat containing all other flats passing through p. By [CCR], M^{n} carries a Cr-structure which is compatible with S. For n = 3, these conclusions hold even if the extra assumptions on F are dropped. Moreover, up to isomorphism, the Cr-structure on M^{3} arising from the construction of [CCR] is independent of the particular nonpositively curved metric.

Original language | English (US) |
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Pages (from-to) | 389-415 |

Number of pages | 27 |

Journal | Communications in Analysis and Geometry |

Volume | 12 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2004 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Geometry and Topology
- Statistics, Probability and Uncertainty

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## Cite this

*Communications in Analysis and Geometry*,

*12*(1-2), 389-415. https://doi.org/10.4310/CAG.2004.v12.n1.a17