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

Let Mⁿ denote a closed Riemannian manifold with nonpositive sectional curvature. Let Xⁿ denote a closed smooth manifold which admits an F- structure, \frak F. If there exists f : Xⁿ → Mⁿ with nonzero degree, then Mⁿ 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 Mⁿ of any flat (i.e. Euclidean slice) of Sis 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 ∈ Mⁿ, there is a flat containing all other flats passing through p. By [CCR], Mⁿ 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 M³ arising from the construction of [CCR] is independent of the particular nonpositively curved metric.

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Published 1 January 2004