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arXiv · 1708.00718

On a rigidity property of perturbations of circle bundles on 3-manifolds

Massimo Villarini
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Abstract

We show that a smooth 1-parameter family of foliations by circles of a closed 3-manifold, deforming the foliation whose leaves are the fibers of a circle bundle, is trivial, i.e. all the foliations of the family arise from circle bundles isomorphic to the unperturbed one, if a continuity property of the Seifert leaves of the perturbation holds true. This rigidity property is true for any real analytic 1-parameter family of foliations by circles when the base space of the circle bundle defining the unperturbed foliation is not a torus. The dimensionality hypothesis is discussed in relation to an example by Thurston of a vector field on a closed 5-manifold whose orbits are closed, with unbounded lenght.

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Massimo Villarini. 2017-08-02. On a rigidity property of perturbations of circle bundles on 3-manifolds. https://arxiv.org/abs/1708.00718

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