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

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Last modified: April 5, 2011