2017/08/02 by Villarini, Massimo
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.00718
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.