Jets of flat partial connections

TL;DR

This paper introduces jets of flat partial connections in the context of smooth foliations, providing new tools to characterize and extend transversely affine and projectively structured foliations, including singular cases.

Contribution

It defines jets of flat partial connections and applies them to characterize and prolong transversely projective structures on foliations of any codimension.

Findings

Characterization of transversely affine structures for codimension one foliations.

Definition of prolongation of transversely projective structures using jets.

Construction of singular transversely projective structures for higher codimension foliations.

Abstract

We define and study jets of flat partial connections in the setting of smooth foliations and flat partial connections on locally free sheaves. In the case of codimension one foliations, we apply this definition to characterize transversely affine and transversely projective structures. For foliations of arbitrary codimension, we use jets of the Bott connection on the normal sheaf to define the prolongation of a transversely projective structure, and then apply it to produce singular transversely projective structures.