### 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) |
---|---|

Pages (from-to) | 389-415 |

Number of pages | 27 |

Journal | Communications in Analysis and Geometry |

Volume | 12 |

Issue number | 1-2 |

State | Published - Jan 2004 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Geometry and Topology

### Cite this

*Communications in Analysis and Geometry*,

*12*(1-2), 389-415.

**Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3.** / Cao, Jianguo; Cheeger, Jeff; Rong, Xiaochun.

Research output: Contribution to journal › Article

*Communications in Analysis and Geometry*, vol. 12, no. 1-2, pp. 389-415.

}

TY - JOUR

T1 - Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3

AU - Cao, Jianguo

AU - Cheeger, Jeff

AU - Rong, Xiaochun

PY - 2004/1

Y1 - 2004/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=4043143496&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4043143496

VL - 12

SP - 389

EP - 415

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

SN - 1019-8385

IS - 1-2

ER -