Global Stability of Distributions of Higher Corank of Derived Length One

Studying a manifold with a certain geometric structure becomes an interesting topic in differential topology when such a structure has some stability. The local stability of symplectic and contact structures, the Darboux theorem, is a fundamental property for symplectic and contact topology. There is a well-known global stability theorem for contact structures, the Gray theorem (see [3]): a one-parameter deformation of a contact structure on a compact orientable manifold can be pursued by using a global isotopy of the manifold. We obtain, in this paper, a theorem of this type for distributions of higher coranks. We consider distributions of corank greater than 1, each of which has the tangent bundle of the underlying manifold as its first derived distribution. Furthermore, comparing the stability of distributions of corank greater than 1 with that of distributions of corank 1, we obtain a generalization of the Gray theorem. In the present paper, we prove the following theorem.