A note on jet and geometric approach to higher order connections

TL;DR

This paper compares geometric and jet-based methods for interpreting higher order connections, analyzing their structures and relationships, with a focus on Ehresmann prolongation and linear connections.

Contribution

It provides a detailed comparison of geometric and jet approaches to higher order connections, highlighting their differences and connections, especially through Ehresmann prolongation.

Findings

Clarifies the relationship between geometric and jet approaches

Analyzes Ehresmann prolongation in the context of higher order connections

Includes discussion on linear connections

Abstract

We compare two ways of interpreting higher order connections. The geometric approach lies in the decomposition of higher order tangent space into the horizontal and vertical structures while the jet--like approach considers a higher order connection as the section of a jet prolongation of a fibered manifold. Particularly, we use the Ehresmann prolongation of a general connection and study the result from the point of view of geometric theory. We pay attention to linear connections, too.