Higher order jet bundels of lie group-valued functions

Abstract

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 Jk(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 Jk(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.