On deformations of transversely homogeneous foliations

A transversely homogeneous foliation is a foliation whose transverse model is a homogeneous space G/H. In this paper we consider the class of transver sely homogeneous foliations F on a manifold M which can be defined by a fam ily of 1-forms on M fulfilling the Maurer-Cartan equation of the Lie group G. This class includes as particular cases Lie foliations and certain homog eneous spaces foliated by points. We develop, for the foliations belonging to this class, a deformation theory for which both the foliation F, and the model homogeneous space G/H are allowed to change. As the main result we s how that, under some cohomological assumptions, there exist a versal space of deformations of finite dimension for the foliations of the class and whe n the manifold M is compact. Some concrete examples are discussed. (C) 2001 Elsevier Science Ltd. All rights reserved.