Higher order jet bundles of Lie group-valued functions

TL;DR

This paper studies the structure of higher order jet bundles of Lie group-valued functions, providing explicit trivializations, groupoid operations, and local expressions using a linear connection.

Contribution

It introduces a method to trivialize jet bundles of Lie group-valued functions and derives explicit formulas for their groupoid operations and local coordinate expressions.

Findings

Explicit trivialization of jet bundles using a linear connection

Formulas for groupoid multiplication and inverse in the trivialized setting

Local coordinate expressions of the trivialization

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 $J^k(X,G)$ 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.