First-order invariants of differential 2-forms

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 $\mathrm{Diff}M$-invariant smooth functions of first order on $O_M$ is proved to be isomorphic to the algebra of smooth $Sp(\hat{I}{\text{c◯}}_x)$-invariant functions on ${\wedge }^3{T}_x^{\ast }M$, $x$ being a fixed point in $M$, and $\hat{I}{\text{c◯}}_x$ a fixed element in $(O_M{)}_x$. The maximum number of functionally independent invariants is computed.