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

Jianguo Cao, Jeff Cheeger, Xiaochun Rong

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)389-415
Number of pages27
JournalCommunications in Analysis and Geometry
Volume12
Issue number1-2
StatePublished - Jan 2004

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F-structure
CR Structure
Closed
Euclidean
Isomorphism
Covering
Denote
Universal Space
Metric
Nonpositive Curvature
Covering Space
Consistency Conditions
Pullback
Smooth Manifold
Sectional Curvature
Fundamental Group
Injective
Slice
Submanifolds
Riemannian Manifold

ASJC Scopus subject areas

  • Mathematics(all)
  • Geometry and Topology

Cite this

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

In: Communications in Analysis and Geometry, Vol. 12, No. 1-2, 01.2004, p. 389-415.

Research output: Contribution to journalArticle

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