On Local Integrability Conditions Of Jet Groupoids

TL;DR

This paper investigates the conditions under which jet groupoids, representing symmetries of geometric objects, are formally integrable, providing a local characterization for transitive cases using their geometric interpretations.

Contribution

It offers a new local characterization of formal integrability for transitive jet groupoids based on their associated geometric objects.

Findings

Provides criteria for formal integrability of jet groupoids

Connects jet groupoid integrability to geometric object properties

Enhances understanding of symmetry structures in differential geometry

Abstract

A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept of formal integrability. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, modelled as a section of a natural bundle. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects.