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Journal of Lie Theory 33 (2023), No. 3, 831--844
Copyright Heldermann Verlag 2023

Higher Order Jet Bundles of Lie Group-Valued Functions

Marco Castrillón López
Facultad de Ciencias Matemáticas, Universidad Complutense, Madrid, Spain

Álvaro Rodríguez Abella
Instituto de Ciencias Matemáticas, Madrid, Spain


For each positive integer $k$, the bundle of $k$-jets of functions from a smooth manifold, $X$, to a Lie group, $G$, is denoted by $J^k(X,G)$ and it is canonically endowed with a Lie groupoid structure over $X$. In this work, we utilize a linear connection to trivialize this bundle, i.e., to build an injective bundle morphism from $J^k(X,G)$ into a vector bundle over $G$. Afterwards, we give the explicit expression of the groupoid multiplication on the trivialized space, as well as the formula for the inverse element. In the last section, a coordinated chart on $X$ is considered and the local expression of the trivialization is computed.

Keywords: Fiber bundle, Lie groupoid, jet bundle, partition, tensor product.

MSC: 22E30, 58A20, 22E60.

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