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# Communications in Analysis and Geometry

## Volume 12 (2004)

### Number 1

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

Pages: 389 – 415

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n1.a17

#### Authors

#### 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, \frak 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, \frak F, *S*, satisfy a consistency condition. If \frak 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 \frak 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.