Editing Connection (mathematics) (section)

==Historical survey of connections==

Historically, connections were studied from an [[infinitesimal]] perspective in [[Riemannian geometry]].  The infinitesimal study of connections began to some extent with [[Elwin Bruno Christoffel|Elwin Christoffel]].   This was later taken up more thoroughly by [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]] {{harv|Levi-Civita|Ricci|1900}} who observed in part that a connection in the infinitesimal sense of Christoffel also allowed for a notion of [[parallel transport]].

The work of Levi-Civita focused exclusively on regarding connections as a kind of [[differential operator]] whose parallel displacements were then the solutions of [[differential equation]]s.  As the twentieth century progressed, [[Élie Cartan]] developed a new notion of connection.  He sought to apply the techniques of [[Pfaffian system]]s to the geometries of [[Felix Klein]]'s [[Erlangen program]].  In these investigations, he found that a certain infinitesimal notion of connection (a [[Cartan connection]]) could be applied to these geometries and more: his connection concept allowed for the presence of [[curvature]] which would otherwise be absent in a classical Klein geometry.  (See, for example, {{harv|Cartan|1926}} and {{harv|Cartan|1983}}.)  Furthermore, using the dynamics of [[Gaston Darboux]], Cartan was able to generalize the notion of parallel transport for his class of infinitesimal connections.  This established another major thread in the theory of connections: that a connection is a certain kind of [[differential form]].

The two threads in connection theory have persisted through the present day: a connection as a differential operator, and a connection as a differential form.  In 1950, [[Jean-Louis Koszul]] {{harv|Koszul|1950}} gave an algebraic framework for regarding a connection as a differential operator by means of the [[Koszul connection]].  The Koszul connection was both more general than that of Levi-Civita, and was easier to work with because it finally was able to eliminate (or at least to hide) the awkward [[Christoffel symbols]] from the connection formalism.  The attendant parallel displacement operations also had natural algebraic interpretations in terms of the connection.  Koszul's definition was subsequently adopted by most of the differential geometry community, since it effectively converted the ''analytic'' correspondence between covariant differentiation and parallel translation to an ''algebraic'' one.

In that same year, [[Charles Ehresmann]] {{harv|Ehresmann|1950}}, a student of Cartan's, presented a variation on the connection as a differential form view in the context of [[principal bundle]]s and, more generally, [[fibre bundle]]s.  [[Ehresmann connection]]s were, strictly speaking, not a generalization of Cartan connections.  Cartan connections were quite rigidly tied to the underlying [[differential topology]] of the manifold because of their relationship with [[Cartan's equivalence method]].  Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such as [[Shiing-Shen Chern]], who had already begun moving away from Cartan connections to study what might be called [[gauge connection]]s.  In Ehresmann's point of view, a connection in a principal bundle consists of a specification of [[vertical bundle|''horizontal'' and ''vertical'' vector fields]] on the total space of the bundle.  A parallel translation is then a lifting of a curve from the base to a curve in the principal bundle which is horizontal.  This viewpoint has proven especially valuable in the study of [[holonomy]].