# First-order invariants of differential 2-forms

## TL;DR

This paper characterizes the algebra of first-order invariants of differential 2-forms on a 2n-dimensional manifold, linking it to symplectic invariants and computing their maximum number.

## Contribution

It establishes an isomorphism between the algebra of invariant functions on the open subbundle and symplectic invariants, providing a new understanding of invariants of differential 2-forms.

## Key findings

- Isomorphism between invariant function algebras and symplectic invariants
- Maximum number of functionally independent invariants computed
- Characterization of invariants for differential 2-forms on manifolds

## Abstract

Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(\Omega_{x})$-invariant functions on $\wedge^{3}T_{x}^{\ast}M$, $x$ being a fixed point in $M$, and $\Omega_{x}$ a fixed element in $(O_{M})_{x}$. The maximum number of functionally independent invariants is computed.