Lie jets and symmetries of prolongations of geometric objects

Abstract

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