Parallel foliations

Abstract

The basic theory of foliations is introduced in Chapter 1. Various classes of affine connexions associated with a foliation are discussed, in particular those which give rise to the notion of parallel foliation and those which give a realisation of the 1-Jet holonomy group of C. Ehresmann. In Chapter 2, locally affine foliations are defined as parallel foliations for which the induced structure on each leaf is flat. A local characterisation is given in terms of the existence of a special sub-atlas of co-ordinate charts. Some results are obtained about the global structure of such foliations when certain completeness assumptions are made. Chapter 3 gives a description, in terms of grid manifolds, of the work of S. Kashiwabara on the reducibility of an affinely connected manifold. The work of the first three chapters is then used in Chapter 4 to discuss the question of parallel foliations on pseudoriemannian manifolds. Some new examples are given. An elementary proof of the De Rham-Wu decomposition theorem and some theorems about null foliations determined by submersions are obtained. Chapter 5 is concerned with the properties of pseudoriemannian manifolds which admit systems of parallel vector fields. The problem is discussed in terms of parallel foliations and some recently developed techniques infoliation theory are used to obtain some strong global structure theorems.