On the axiomatization of "parallel transports" and and one its generalization

TL;DR

This paper explores the axiomatic foundations of parallel transport, establishes a bijection with connections, and introduces transports along paths as a generalization, discussing their properties.

Contribution

It provides an axiomatic framework linking connections and parallel transports and generalizes the concept to transports along paths.

Findings

Established a bijective correspondence between connections and parallel transports

Introduced and analyzed properties of transports along paths

Extended the concept of parallel transport to a more general framework

Abstract

A concise discussion of the axiomatic approach to the concept of parallel transport is presented. Attention is drawn to a bijective map between the sets of connections and (axiomatically defined) parallel transports. The transports along paths are pointed as a generalization of the (axiomatically defined) parallel transports. Some properties of the general transports along paths are presented.