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2 papers
Lie jets and symmetries of prolongations of geometric objects
V. V. Shurygin Kazan State University
Abstract:
The Lie jet
$\mathcal L_\theta\lambda$ of a field of geometric objects
$\lambda$ on a smooth manifold
$M$ with respect to a field
$\theta$ of Weil
$\mathbf A$-velocities is a generalization of the Lie derivative
$\mathcal L_v\lambda$ of a field
$\lambda$ with respect to a vector field
$v$. In this paper, Lie jets
$\mathcal L_\theta\lambda$ are applied to the study of
$\mathbf A$-smooth diffeomorphisms on a Weil bundle
$T^\mathbf AM$ of a smooth manifold
$M$, which are symmetries of prolongations of geometric objects from
$M$ to
$T^\mathbf AM$. It is shown that vanishing of a Lie jet
$\mathcal L_\theta\lambda$ is a necessary and sufficient condition for the prolongation
$\lambda^\mathbf A$ of a field of geometric objects
$\lambda$ to be invariant with respect to the transformation of the Weil bundle
$T^\mathbf AM$ induced by the field
$\theta$. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle
$T^2M$ are considered in more detail.
UDC:
514.76
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