Editing Connection (mathematics) (section)

==Possible approaches==

*A rather direct approach is to specify how a [[covariant derivative]] acts on elements of the [[module (mathematics)|module]] of [[vector field]]s as a [[differential operator]].  More generally, a similar approach applies for [[connection (vector bundle)|connections]] in any [[vector bundle]].
*Traditional index notation specifies the connection by components; see [[Christoffel symbols]].  (''Note'': this has three indices, but is '''''not''''' a [[tensor]]).
*In [[pseudo-Riemannian]] and [[Riemannian geometry]] the [[Levi-Civita connection]] is a special connection associated to the [[metric tensor]].
*These are examples of [[affine connection]]s. There is also a concept of [[projective connection]], of which the [[Schwarzian derivative]] in [[complex analysis]] is an instance.  More generally, both affine and projective connections are types of [[Cartan connection]]s.
*Using [[principal bundle]]s, a connection can be realized as a [[Lie algebra]]-valued [[differential form]].  See [[connection (principal bundle)]].
*An approach to connections which makes direct use of the notion of transport of "data" (whatever that may be) is the [[Ehresmann connection]].
*The most abstract approach may be that suggested by [[Alexander Grothendieck]], where a [[Grothendieck connection]] is seen as [[descent (category theory)|descent]] data from infinitesimal neighbourhoods of the [[diagonal]]; see {{harv|Osserman|2004}}.