Differential Geometric Structures and Integrability

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.