Estimates of the action order of some jet groups

Karolina Lubas and Andrzej Zajtz


Abstract. As we know, if the general $k$-jet group $G(n,k)$ in $\R$ acts continuously on a manifold of finite dimension $m>0$ and the highest $k$-th order derivatives act nontrivially, then $k$ is bounded by a known function of $m$ and $n$ [4]. In geometric interpretation the estimate function translates into the order of the natural bundle in which the group $G(n,k)$ acts on the standard fibre (meant as the space of geometric objects). On manifolds with extra structure the same question arises for corresponding subgroups of $G(n,k)$, preserving the structure. In this note the sharp upper estimates are found for actions of unimodular, homothetic and symplectic subgroups.

AMSclassification. 58G99

Keywords. Unimodular, homothetic, symplectic jet groups and Lie algebras, bounds to codimension of subalgebras, action order