Lie jets and symmetries of prolongations of geometric objects

Abstract

The Lie jet Lθλ 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 Lvλ of a field λ with respect to a vector field v. In this paper, Lie jets Lθλ are applied to the study of A-smooth diffeomorphisms on a Weil bundle TAM of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to TAM. It is shown that vanishing of a Lie jet Lθλ is a necessary and sufficient condition for the prolongation λA of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle TAM induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T2M are considered in more detail. © 2011 Springer Science+Business Media, Inc.