Differential Geometry of Foliations

The purpose of this chapter is to introduce the notion of foliation as a particular and fundamental example of a differential geometric structure. Thus, the first step is to introduce various approaches to the definition of a structure, and show how the principal examples, especially foliations, fit in. Next, since a foliation can be viewed as an integrable reduction of the group of the tangent bandle, some general facts about integrability are introduced. As an illustration, the case where the coordinate changes are translations in euclidean space is worked out in detail. Then other important examples, especially foliations, are discussed. In this discussion, the case of only finitely many derivatives is considered carefully, since the difference among orders of differentiability is of increasing geometric significance. Finally, some concepts of particular usefulness in foliation theory are studied and a variety of examples given as motivation for later chapters.

One aim of this chapter is to present some standard facts about connections and their relation to characteristic classes, and then show how these ideas extend to give new kinds of characteristic classes associated to a foliation. The presentation of the material will not be at all standard, however. Indeed, the same characteristic classes associated to a foliation of codimension q can be obtained from the algebra of formal power series vector fields at the origin in ℝ ^q , without any reference to foliations at all. This result is consistent with the point of view that foliations are the natural quotient objects in differential geometry, but it also suggests an exposition in which higher order derivatives are incorporated from the beginning, in order to make the relation appear as clear and natural as possible. Thus, the first section is a discussion of the Lie groups of invertible polynomial mappings. The composition of two mappings of degree k is made into a mapping of degree k by discarding higher order terms, a procedure consistent with the application to Taylor series of differentiable mappings discussed in the second section. The higher order frame bundles on a manifold are principal bundles whose groups are truncated polynomial groups. In addition, each carries a Lie algebra valued 1-form, the tautological form, which contains precisely the information that the derivatives of various orders are not independent when more than one point is involved. Any G-structure has associated to it reductions of the higher order frame bundles to subgroups of the truncated polynomial groups.

In the second chapter, certain cohomology classes have been defined on a foliated manifold. These are obtained by using differential forms related to the characteristic classes of the normal bundle. Moreover, the cohomology classes have formal properties analogous to those of the characteristic classes of a vector bundle. Hence, it is reasonable to try to obtain them by mapping into a classifying space for foliations, that is, some sort of foliated space whose cohomology consists of the characteristic classes of its foliation, and such that all foliations are obtained (up to some reasonable equivalence) by mapping into the classifying space and pulling back its foliation. Even for vector bundles, the classifying space is not a finite dimensional smooth manifold, so it is necessary to generalize the definition to allow foliations of much more general topological spaces, and so that the generalized foliations map contravariantly under a much larger class of mappings. In particular, even for a smooth manifold some class of singularities must be allowed. In the three sections of this chapter, three types of classifying spaces and the accompanying notions of singularity will be discussed.

The first three chapters give background material and relations between foliations and other differential geometric structures. Though current research and open problems are discussed, some of the latter do not seem likely to be solved immediately. In this chapter, the topics discussed are the heart of current activity in the field.