[Paper Review] Transports along paths in fibre bundles. II. Ties with the theory of connections and parallel transports

Bozhidar Z. Iliev|ArXiv.org|Mar 1, 2005

Optical Network Technologies27 references3 citations

TL;DR

This paper establishes that parallel transport in fibre bundles is a special case of transports along paths, unifying the theory of connections with the broader framework of path-dependent transports. It proves that parallel transports—whether connection-generated or axiomatically defined—satisfy additional invariance conditions under reparametrization, thereby embedding them within the general theory of transports along paths.

ABSTRACT

A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the former.

Motivation & Objective

• To clarify the relationship between transports along paths and parallel transports in fibre bundles.
• To demonstrate that parallel transports are a special class of transports along paths satisfying additional invariance conditions.
• To unify the connection-based and axiomatic approaches to parallel transport under the broader framework of path transports.
• To investigate the role of reparametrization invariance in linear transports along paths, particularly in vector bundles.
• To show how linear transports generated by derivations of tensor algebras can be interpreted as parallel transports under the axiomatic definition.

Proposed method

• Formalizing transports along paths as maps $ I^{ au}_{s\to t} : \pi^{-1}(\gamma(s)) \to \pi^{-1}(\gamma(t)) $ satisfying composition and identity axioms.
• Introducing the key condition (1.6): $ I^{\gamma \circ \tau}_{s\to t} = I^{\gamma}_{\tau(s)\to \tau(t)} $, which ensures reparametrization invariance.
• Proving that any parallel transport—whether connection-generated or axiomatically defined—satisfies this invariance condition.
• Using matrix representations $ H(s,t;\gamma) $ and connection coefficients $ \Gamma_{\gamma}(s) $ to characterize linear transports along paths.
• Establishing equivalence between reparametrization invariance and the differential equation $ \Gamma_{\gamma \circ \tau}(s) = \frac{d\tau(s)}{ds} \cdot \Gamma_{\gamma}(\tau(s)) $.
• Relating the transport to the covariant derivative $ \mathcal{D}^\gamma_s $ and showing that invariance implies $ \mathcal{D}^{\gamma \circ \tau}_s = \frac{d\tau(s)}{ds} \cdot \mathcal{D}^{\gamma}_{\tau(s)} $.

Experimental results

Research questions

• RQ1 How are parallel transports in fibre bundles related to the general theory of transports along paths?
• RQ2 What additional conditions must a transport along paths satisfy to qualify as a parallel transport?
• RQ3 Can parallel transports defined axiomatically be embedded within the framework of transports along paths?
• RQ4 What is the role of reparametrization invariance in linear transports along paths in vector bundles?
• RQ5 How do the matrix representation $ H(s,t;\gamma) $, connection coefficients $ \Gamma_{\gamma}(s) $, and covariant derivative $ \mathcal{D}^\gamma_s $ relate under reparametrization?

Key findings

• Any parallel transport, whether generated by a connection or defined axiomatically, is a transport along paths satisfying the reparametrization invariance condition (1.6).
• The condition (1.6) ensures that the transport depends only on the image curve, not on the parametrization of the path.
• For linear transports in vector bundles, reparametrization invariance is equivalent to the differential equation $ \Gamma_{\gamma \circ \tau}(s) = \frac{d\tau(s)}{ds} \cdot \Gamma_{\gamma}(\tau(s)) $.
• The equivalence between (5.1), (5.2), (5.3), and (5.4) confirms that reparametrization invariance is preserved under matrix, coefficient, and derivative representations.
• The result shows that the theory of transports along paths is a general framework that includes parallel transport as a special case.
• Wilson loops in gauge theory are shown to be consistent with this framework, as they represent parallel transport along closed loops and are invariant under reparametrization.