Differential forms on the Fréchet manifold $ℱ(S,M)$ of smooth functions on a compact $k$-dimensional manifold $S$ can be obtained in a natural way from pairs of differential forms on $M$ and $S$ by the hat pairing. Special cases are the transgression map ${\Omega }^p\left(M\right)\to {\Omega }^{p-k}\left(ℱ(S,M)\right)$ (hat pairing with a constant function) and the bar map ${\Omega }^p\left(M\right)\to {\Omega }^p\left(ℱ(S,M)\right)$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6].

Soit $\omega$ un germe en $0\in {\mathbf{C}}^n$ de 1-forme différentielle holomorphe vérifiant la condition d’intégrabilité $\omega \wedge d\omega =0$. S’il existe un germe $h$ d’application holomorphe de $({\mathbf{C}}^r,0)$ dans $({\mathbf{C}}^n,0)$ qui possède les deux propriétés suivantes :a) $h^{*}\left(\omega \right)$ a une intégrale première formelle,b) la codimension du lieu singulier $S\left(h^{*}\right(\omega \left)\right)$ de $h^{*}\left(\omega \right)$ est supérieure ou égale à 2,alors $\omega$ a une intégrale première holomorphe.

In this work, we consider variational problems defined by $G$-invariant Lagrangians on the $r$-jet prolongation of a principal bundle $P$, where $G$ is the structure group of $P$. These problems can be also considered as defined on the associated bundle of the $r$-th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.

We study the special Lagrangian Grassmannian $SU\left(n\right)/SO\left(n\right)$, with $n\ge 3$, and its reduced space, the reduced Lagrangian Grassmannian $X$. The latter is an irreducible symmetric space of rank $n-1$ and is the quotient of the Grassmannian $SU\left(n\right)/SO\left(n\right)$ under the action of a cyclic group of isometries of order $n$. The main result of this paper asserts that the symmetric space $X$ possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank $\ge 2$, which is...