Jets of flat partial connections

TL;DR

This work develops a theory of jets of flat partial connections on foliations, pairing jets with flatness to study transversal geometric structures. It characterizes transversely affine and transversely projective structures for codimension one foliations via first- and second-order transverse differential equations, and constructs a prolongation mechanism that produces singular transversely projective structures using the Bott connection on the normal bundle. The framework leverages jet bundles, Cartan-like structures, and Schwarzian calculus to relate differential equations on foliations to transverse geometries, extending classical Deligne–Gunning viewpoints to singular settings. The prolongation results additionally provide a pathway to generate and study higher-order transverse structures on foliations through jet-theoretic and group-action techniques, with potential applications to residue theories and singular transverse geometries.

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.

Theorems & Definitions (65)

• Remark 2.1
• Remark 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Proposition 3.6
• Lemma 3.7
• proof
• proof : Proof of Proposition \ref{['P: flat connection and existence of flat basis']}