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Uniformly quasi-isometric foliations. (English) Zbl 0784.58060

A pseudogroup \({\mathcal H}\) of local \(C^ 1\)-diffeomorphisms acting on a Riemannian manifold \(M\) is called uniformly quasi-isometric if there is a constant \(A \geq 1\) such that the inequality \(| v |/A \leq | df(v)| \leq A | v |\) is satisfied for all \(f \in {\mathcal H}\), \(v\in T_ xM\), \(x \in \text{dom} (f)\), where \(| \cdot |\) is the length of tangent vectors.
The paper is concerned with \(C^ 1\)-foliations \({\mathcal F}\) of general codimension \(q\) on a compact Riemannian manifold \(N\) such that the associated infinitesimal holonomy pseudogroups acting on \(\mathbb{R}^ q\) are uniformly quasi-isometric. Main result: the closure \(\overline F\) of every leaf \(F\) is a \(C^ 0\)-imbedded submanifold, moreover, the closure \(\overline{\mathcal H}\) of the holonomy group provides locally transitive actions on \(\overline F\). The proof is based on several recent improvements of famous Montgomery-Gleason solution of Hilbert’s fifth problem. Thorough introduction of fundamental concepts, complete proofs, and precise references made the article self-contained. Interesting historical comments and several unsolved problems are involved.
Reviewer: J.Chrastina (Brno)

MSC:

58H05 Pseudogroups and differentiable groupoids
57R30 Foliations in differential topology; geometric theory
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