Affine connections and symmetry jets

TL;DR

This paper establishes a bijective link between affine connections and symmetry jets, providing a geometric framework to describe connections, their properties, and related tensors using groupoid morphisms and jets.

Contribution

It introduces symmetry jets as a new way to characterize affine connections and develops a geometric, intrinsic description of connection-related objects via groupoid morphisms.

Findings

Affine connections correspond bijectively to symmetry jets.

Torsion-free connections are characterized by geodesic symmetries.

Obstructions to holonomicity relate to torsion, curvature, and their derivatives.

Abstract

We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of $2$-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descriptions of various related objects involving the gauge groupoid of the frame bundle. In particular, the property of uniqueness of affine extension admits an equivalent formulation as the property of existence and uniqueness of a certain groupoid morphism. Moreover, affine extension may be carried out at all orders and this allows for a description of the tensors associated to an affine connections, namely the torsion, the curvature and their covariant derivatives of all orders, as obstructions for the affine extension to be holonomic. In addition this framework provides a nice interpretation for the absence of other tensors.