Codimension one minimal foliations and the fundamental groups of leaves. (English) Zbl 1148.53017

Yokoyama, Tomoo; Tsuboi, Takashi

Ann. Inst. Fourier 58, No. 2, 723-731 (2008).

The authors study codimension one, transversely orientable, real-analytic foliations of a paracompact manifold for which every leaf is dense. Under these conditions, they prove that the foliation is without holonomy if either (1) the fundamental group of every leaf is isomorphic to \({\mathbb Z}\) or (2) the 2nd homotopy group of the ambient manifold is zero and the fundamental group of every leaf is isomorphic to \({\mathbb Z}^k\) for some \(k \geq 0\). They remark that the assumption of real-analyticity is there solely to guarantee that there are no null homotopic closed transverse curves to the foliation. An example is given to show the necessity of the denseness of the leaves. They also construct an example of a foliation with nontrivial holonomy with diffeomorphic leaves.

Reviewer: James Hebda (St. Louis)

MSC:

53C12	Foliations (differential geometric aspects)
57R30	Foliations in differential topology; geometric theory

Keywords:

foliations; real-analytic; holonomy; fundamental group of leaves

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