Skew-symmetric differential forms. Invariants. Realization of invariant structures

Abstract

Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was introduced by E.Cartan for a notation of integrand expressions, which can create the integral invariants.(The existence of integral invariants was recognized by A.Poincare while studying the general equations of dynamics.) All invariant mathematical formalisms are based on invariant properties of closed exterior forms. The invariant properties of closed exterior forms explicitly or implicitly manifest themselves essentially in all formalisms of field theory, such as the Hamilton formalism, tensor approaches, group methods, quantum mechanics equations, the Yang-Mills theory and others. They lie at the basis of field theory. However, in this case the question of how the closed exterior forms are obtained arises. In present work it is shown that closed exterior forms, which possess the invariant properties, are obtained from skew-symmetric differential forms, which, as contrasted to exterior forms, are defined on nonintegrable manifolds. The process of generating closed exterior forms describes the mechanism of realization of invariants and invariant structures.