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 $$**A**-smooth diffeomorphisms on a Weil bundle
$T$**A**M of a smooth manifold $M$, which are symmetries of
prolongations of geometric objects from $M$ to $T$**A**M.
It is shown that vanishing of a Lie jet $L$_{q}
l is
a necessary and sufficient condition for the prolongation
$$l^{A}
of a field of geometric
objects $$l to be invariant with
respect to the transformation of the Weil bundle $T$**A**M induced by
the field $$q.
The case of symmetries of prolongations of fields of geometric objects
to the second-order tangent bundle $T2M$ are
considered in more detail.

Location: http://mech.math.msu.su/~fpm/eng/k10/k102/k10217h.htm

Last modified: April 5, 2011