Connection of the differential-geometrical structures with   skew-symmetric differential forms. Forming differential-geometrical   structures and manifolds
  The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures.   The using of skew-symmetric differential forms enables one to reveal a mechanism of forming differential-geometrical structures as well. To do this, it is necessary to consider the skew-symmetric differential forms whose basis, unlike to the case of exterior differential forms, are manifolds with unclosed metric forms. Such differential forms possess the evolutionary properties. They can generate closed differential forms, which correspond to the differential-geometrical structures. 
L.I. Petrova