Codimension one Foliations of Closed Manifolds

Abstract

The global solutions (integral manifolds) of a completely integrable Pfaffian system are in general very intricately interwoven. Indeed, the appropriate language with which to describe a given foliation (i.e., the family of solutions of such a system) has not been fully developed, since the range of possibilites is not entirely known. This paper provides part of the information needed for a global classification, up to a diffeomorphism preserving the foliation, in the case where the n-manifold M on which the system (equation) is defined is compact without boundary and the dimension of the integral manifolds (leaves) is (n - 1). Euler begins the third volume of his Institutionum Calculi Integralis [3] with a treatment of global singular codimension one foliations on R3. His starting point is that when certain one-forms =Pdx + Qdy + Rdz are multiplied by functions M they become exact differentials d V, and their complete integrals are obtained by setting V equal to constants. He first observes that Mdcv - (v A dM 0, and then eliminates M and its derivatives by computing GO A do 0. This latter condition, written explicitly as