A multilagrangian fiber bundle is a fiber bundle $P$ over an $n$ -dimensional manifold $M$ equipped with a $(k+1-r)$ -horizontal $(k+1)$ -form $\,\omega$ of constant rank on the total space $P$ , where $\,1\leqslant r\leqslant k+1\,$ and $\,k+1-r\leqslant n$ , called the multilagrangian form and said to be of rank $N$ and horizontality degree $k+1-r$ , such that $\,\omega$ is closed,

and such that at every point $p$ of $P$ , $\omega_{p}$ is a multilagrangian form of rank $N$ on the tangent space $T_{p}P$ . If the multilagrangian subspaces at the different points of $P$ fit together into a distribution $L$ on $P$ (which is contained in the vertical bundle $VP$ of $P$ ), we call it the multilagrangian distribution of $\,\omega$ .