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



Jianguo Cao

Jeff Cheeger

Xiaochun Rong


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 : XnMn 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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