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

TL;DR

This paper proves a rigidity property for smooth deformations of circle bundle foliations on 3-manifolds, showing such perturbations are trivial under certain continuity conditions, especially when the base space is not a torus.

Contribution

It establishes a rigidity theorem for perturbations of circle bundle foliations on 3-manifolds, extending to real analytic families and discussing dimensionality constraints.

Findings

Deformations preserving leaf continuity are trivial, maintaining the original circle bundle structure.

The rigidity holds for real analytic families when the base space is not a torus.

Discusses the role of dimensionality through Thurston's example of vector fields on 5-manifolds.

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.