The Classification Problem for 2-Forms in Four Variables

TL;DR

This paper introduces a classification framework for differential 2-forms in four variables, providing normal models for forms of type less than 4 and reducing the problem for type 4 forms to symplectic linear frame equivalence.

Contribution

It defines the notion of type for 2-forms in four variables and solves the local equivalence problem for generic cases by linking it to symplectic linear frame equivalence.

Findings

Normal models for 2-forms of type < 4

Reduction of type 4 forms to symplectic linear frame equivalence

Solution to the local equivalence problem for generic 2-forms

Abstract

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $\Omega$ is 4, then the equivalence under diffeomorphisms of $\Omega$ is reduced to the equivalence of a symplectic linear frame functorially attached to $\Omega$. As the equivalence problem for linear parallelisms is known, the present work solves generically the equivalence problem under diffeomorphisms of germs of 2-forms in 4 variables.