FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 2, PAGES 163-181

Lie jets and symmetries of prolongations of geometric objects

V. V. Shurygin

Abstract

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The Lie jet Lq l of a field of geometric objects l on a smooth manifold M with respect to a field q of Weil A-velocities is a generalization of the Lie derivative Lv l of a field l with respect to a vector field v. In this paper, Lie jets Lq l 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 Lq l is a necessary and sufficient condition for the prolongation lA of a field of geometric objects l to be invariant with respect to the transformation of the Weil bundle TAM induced by the field q. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T2M are considered in more detail.

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