Evolutionary Forms: the Generation of Differential-geometrical Structures. (symmetries and Conservation Laws.)

Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (manifolds with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closure conditions of inexact exterior form (vanishing the differentials of exterior and dual forms) point out to the fact that the closed inexact exterior form is a quantity conserved on pseudostructure having the dual form as the metric form. We obtain that the closed inexact exterior form and corresponding dual form made up a conservative object, i.e. a quantity conserved on pseudostructure. Such conservative object corresponds to the conservation law and is a differential-geometrical structure. Transition from the evolutionary form to the closed inexact exterior form describes the process of generating the differential-geometrical structures. This transition is possible only as a degenerate transformation, the condition of which is a realization of a certain symmetry. Physical structures that made up physical fields are such differential-geometrical structures. And they are generated by material systems (medias). Relevant symmetries are caused by the degrees of freedom of material system. Exterior and evolutionary forms. The difference between exterior and evolutionary skew-symmetric differential forms is connected with the properties of manifolds on which skew-symmetric forms are defined. It is known that the exterior differential forms [1] are skew-symmetric differential forms whose basis are differentiable manifolds or they can be manifolds with structures of any type. Structures, on which exterior forms are defined, have closed metric forms. It has been named as evolutionary forms skew-symmetrical differential forms whose basis are deforming manifolds, i.e. manifolds with unclosed metric forms. The metric form differential, and correspondingly its commutator, are nonzero. The commutators of metric forms of such manifolds describe a manifold deformation (torsion, curvature and so on). Lagrangian manifolds, manifolds constructed of trajectories of material system elements, tangent manifolds of differential equations describing physical processes and others can be examples of deforming manifolds. A specific feature of the evolutionary forms, i.e skew-symmetric forms defined on deforming manifolds, is the fact that commutators of these forms include commutators of the manifold metric forms being nonzero. Such commutators 1