Lie jets and symmetries of prolongations of geometric objects

The Lie jet $\mathcal L_\theta\lambda$ of a field of geometric objects $\lambda$ on a smooth manifold $M$ with respect to a field $\theta$ of Weil $\mathbf A$-velocities is a generalization of the Lie derivative $\mathcal L_v\lambda$ of a field $\lambda$ with respect to a vector field $v$. In this paper, Lie jets $\mathcal L_\theta\lambda$ are applied to the study of $\mathbf A$-smooth diffeomorphisms on a Weil bundle $T^\mathbf AM$ of a smooth manifold $M$, which are symmetries of prolongations of geometric objects from $M$ to $T^\mathbf AM$. It is shown that vanishing of a Lie jet $\mathcal L_\theta\lambda$ is a necessary and sufficient condition for the prolongation $\lambda^\mathbf A$ of a field of geometric objects $\lambda$ to be invariant with respect to the transformation of the Weil bundle $T^\mathbf AM$ induced by the field $\theta$. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle $T^2M$ are considered in more detail.