1-forms with the exterior derivative of maximal rank

Abstract

On a manifold M let ω be a 1-form, whose exterior derivative dω has the rank=dim M at every point of Λf. Roughly speaking, this condition on ω means that locally ω cannot be obtained by pulling back any 1-form from a lower dimensional manifold (E. Cartan [4]). The existence of ω on M sets some limits to the topology and the manifold structure of M. Clearly M must be even-dimensional and orientable. M cannot be compact (§5). In this paper we will obtain more information in this regard. In particular we will prove (Theorem 4.1) that M is the cotangent bundle of another manifold 5, if M satisfies certain completeness conditions with respect to ω, and the singularities of ω have certain properties. Clearly some completeness is necessary to characterize a cotangent bundle. We would, however, like to emphasize the importance of the singularities in the study of M with α>. A point x of M is called singular if ω vanishes at x. Let 5 denote the set of all singular points on M. Locally the complement M — S with ω is trivial; M — 5 with ω is locally m isomorphic to R with £ xdx+ (see E. Cartan [4, p. 264], and also Arens ΐ = l