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\(R\)-covered foliations of hyperbolic 3-manifolds. (English) Zbl 0924.57014

A foliation \({\mathcal F}\) of a compact 3-manifold \(M\) is \(\mathbb{R}\)-covered if the pulled-back foliation \(\widetilde{\mathcal F}\) of the universal cover \(\widetilde M\) of \(M\) is topologically the standard foliation of \(\mathbb{R}^3\) by horizontal \(\mathbb{R}^2\)’s. The foliation \({\mathcal F}\) is uniform if in the foliation \(\widetilde{\mathcal F}\) every two leaves are a bounded distance apart. The author studies: \(\mathbb{R}\)-covered foliations that are not uniform, Lorentz cone fields, general \(\mathbb{R}\)-covered foliations and the instability of \(\mathbb{R}\)-covered foliations.
Theorem. There exist foliations of hyperbolic 3-manifolds which are \(\mathbb{R}\)-covered but not uniform. They can be chosen arbitrarily close to surface bundles over circles.
A hyperbolic example is considered. The method uses transverse regulating vector fields and cone fields for \(\mathbb{R}\)-covered foliations.

MSC:

57M50 General geometric structures on low-dimensional manifolds
53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
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References:

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