On the history of Levi-Civita's parallel transport

TL;DR

This paper explores the historical development of Levi-Civita's concept of parallel transport, emphasizing the role of the principle of virtual work in its formulation and its impact on the development of connections and gauge theories.

Contribution

It provides a detailed historical and technical reconstruction of Levi-Civita's original argument and discusses its significance in the evolution of differential geometry and physics.

Findings

Highlighting the role of virtual work in defining parallel transport

Reconstructing Levi-Civita's original argument in detail

Discussing the impact on gauge theories and connections

Abstract

In this historical note, we wish to highlight the crucial conceptual role played by the principle of virtual work of analytical mechanics, in working out the fundamental notion of parallel transport on a Riemannian manifold, which opened the way to the theory of connections and gauge theories. Moreover, after a detailed historical-technical reconstruction of the original Levi-Civita's argument, a further historiographical deepening and a related critical discussion of the question, are pursued.