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

TL;DR

This paper explores how closed exterior skew-symmetric differential forms, which are invariants in mathematics and physics, are generated from skew-symmetric forms on nonintegrable manifolds, elucidating the realization of invariant structures.

Contribution

It demonstrates that closed exterior forms are derived from skew-symmetric forms on nonintegrable manifolds, revealing the mechanism behind the realization of invariants and invariant structures.

Findings

Closed exterior forms are obtained from skew-symmetric forms on nonintegrable manifolds.

The process describes how invariants and invariant structures are realized.

Invariant properties of closed exterior forms underpin many field theory formalisms.

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.