# Filip Petrák

## Higher-order and Weil Grassmannian as orbit spaces

We define a foliation on $reg J^{r}_{0}(Rk, Rm)$ with $G^{r}_{m}$-orbits as its leaves and the left action given by the jet composition. We construct an atlas on $Gr(r, k, m)$ from the finite system of local sections of ˆp# identified with local maps with supports dense in $V = Gr(r, k, m)$. In the present work, we investigate the case of $m<k$. The symbol $reg$ indicates submersions, the jet group acts from the left and in the most general situation corresponding to a Weil functor $T^{A}$ we do not work with the group $Aut A$ but still with $G^{r}_{m}$.