Complete Foliations with Transverse Rigid Geometries and Their Basic Automorphisms

Abstract

The notion of rigid geometry is introduced. Rigid geometries include Cartan geometries as
well as rigid geometric structures in the sense of Gromov. Foliations with transverse
rigid geometries are investigated. An invariant g0 of a foliation with transverse rigid
geometry, being a Lie algebra, is introduced. We prove that if, for some foliation with
transverse rigid geometry, g0 is zero, then there exists a unique Lie group structure on its full
basic automorphism group. Some estimates of the dimensions of this group depending on the
transverse geometry are obtained. Examples, illustrating the main results, are constructed.