### History of Understanding Bundles with Connection using Parallel Transport around Loops

#### Posted by Urs Schreiber

In the process of finishing a paper, I was today busy collecting some background history literature on the development of the idea that a principal $G$-bundle with connection may equivalently be encoded in its parallel transport around based loops.

Apparently one of the oldest occurrences of the idea that a bundle $P \to X$ with connection may be reconstructed from its holonomies for any fixed base point in the connected base space $X$ appears in

S. Kobayashi,

Comptes Rendus, 238, 443-444. (1954).

(full text)

A more detailed and more general discussion, has then been given in

Milnor, J.

Annals of Mathematics, 63, 272-284.(1956).

A detailed discussion of the differentiable case appears is

C. Teleman
*Généralisation du groupe fondamental*

Annales Scientifiques de l’école Normale Supérieure 3, 77, 195-234.
(1960)

(full text)

C. Teleman

Annali di Matematica, Pura ed Applicata, LXII, 379-412. (1963).

This history is recollected in the introduction of

J. W. Barrett
*Holonomy and path structures in general relativity and Yang-Mills theory*

Int. J. of Theor. Phys., Iiol. 30, No. 9, 1991

(pdf)

who himself gives a proof.

Barret implicitly uses the diffeological – or Chen – smooth structure on the space of paths (except that he restricts attention to closed paths).

In

Jerzy Lewandowski
*Group of loops, holonomy maps, path bundle and path
connection*

Class. Quantum Grav. 10 (1993) 879-904

(pdf)

the statement of the equivalence is attributed to

J. Anandan

*Holonomy groups in gavity and gauge fields*
Pmc. Conf Differential Geometric Methods in Plystcs (fieste 1981)

ed G Denardo and H D Doebner (Singapore: World Scientific)

Therein it is shown that smoothness of the parallel transport is a necessary condition for it to come from a smooth bundle with connection. Barrett also shows that this is sufficient.

Lewandowski adds to this a formulation of an equivalence of bundles with connections and the subset of loops around which the corresponding parallel transport is trivial.

He also comments on the relation to generalized connections (those functors from paths to the group which are required neither to be smooth nor, in fact, continuous) in section 3. (I mentioned this issue here.)

Around the same time appeared

A. Caetano, R.F. Picken
*An axiomatic definition of holonomy*

Int. Journ. Math. 5 (1994) 835

(scanned version).

These authors introduced the idea of *sitting instants* of paths
and noticed that the most elegant way to (re)state the maximal equivalence relation on paths
which is respected by parallel transport is in terms of *thin homotopy*, i.e. smooth homotopy
between paths with differential of less than maximal rank.

Barrett originally had something very similar but slightly different. With Caetano and Picken’s relation, the space of thin homotopy classes of paths in $X$ becomes an honest groupoid $P_1(X)$ internal to smooth spaces.

There would be more to say. But I have to run now, unfortunately (same problem every Friday…)

I am grateful to Christian Fleischhack and to Laurent Guillopé for help with tracking down some of the above links.

## Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Surely this story should start with Riemann and the early days of complex analysis.