Editing Affine connection (section)

==Parallel transport for affine connections==

{{see also|Parallel transport}}

[[File:Parallel transport sphere2.svg|thumb|right|Parallel transport of a tangent vector along a curve in the sphere.]]
Comparison of tangent vectors at different points on a manifold is generally not a well-defined process. An affine connection provides one way to remedy this using the notion of [[parallel transport]], and indeed this can be used to give a definition of an affine connection.

Let {{mvar|M}} be a manifold with an affine connection {{math|∇}}. Then a vector field {{mvar|X}} is said to be '''parallel''' if {{math|∇''X'' {{=}} 0}} in the sense that for any vector field {{mvar|Y}}, {{math|∇<sub>''Y''</sub>''X'' {{=}} 0}}. Intuitively speaking, parallel vectors have ''all their [[derivative]]s equal to zero'' and are therefore in some sense ''constant''. By evaluating a parallel vector field at two points {{mvar|x}} and {{mvar|y}}, an identification between a tangent vector at {{mvar|x}} and one at {{mvar|y}} is obtained. Such tangent vectors are said to be '''parallel transports''' of each other.

Nonzero parallel vector fields do not, in general, exist, because the equation {{math|∇''X'' {{=}} 0}} is a [[partial differential equation]] which is [[overdetermined system|overdetermined]]: the [[integrability condition]] for this equation is the vanishing of the '''curvature''' of {{math|∇}} (see below). However, if this equation is restricted to a [[curve]] from {{mvar|x}} to {{mvar|y}} it becomes an [[ordinary differential equation]]. There is then a unique solution for any initial value of {{mvar|X}} at {{mvar|x}}.

More precisely, if {{math|''γ'' : ''I'' → ''M''}} a [[curve|smooth curve]] parametrized by an interval {{math|[''a'', ''b'']}} and {{math|''ξ'' ∈ T<sub>''x''</sub>''M''}}, where {{math|''x'' {{=}} ''γ''(''a'')}}, then a [[vector field]] {{mvar|X}} along {{mvar|γ}} (and in particular, the value of this vector field at {{math|''y'' {{=}} ''γ''(''b'')}}) is called the '''parallel transport of {{mvar|ξ}} along {{mvar|γ}}''' if
#{{math|∇<sub>''γ′''(''t'')</sub>''X'' {{=}} 0}}, for all {{math|''t'' ∈ [''a'', ''b'']}}
#{{math|''X''<sub>''γ''(''a'')</sub> {{=}} ''ξ''}}.
Formally, the first condition means that {{mvar|X}} is parallel with respect to the [[pullback (differential geometry)|pullback connection]] on the [[pullback bundle]] {{math|''γ''<sup>∗</sup>T''M''}}. However, in a [[local trivialization]] it is a first-order system of [[linear differential equation|linear ordinary differential equations]], which has a unique solution for any initial condition given by the second condition (for instance, by the [[Picard–Lindelöf theorem]]).

Thus parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a [[linear isomorphism]] between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general depend on the choice of the curve: if it does not, then parallel transport along every curve can be used to define parallel vector fields on {{mvar|M}}, which can only happen if the curvature of {{math|∇}} is zero.

A linear isomorphism is determined by its action on an [[Basis (linear algebra)#Ordered bases and coordinates|ordered basis]] or '''frame'''. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) [[frame bundle]] {{math|GL(''M'')}} along a curve. In other words, the affine connection provides a '''lift''' of any curve {{mvar|γ}} in {{mvar|M}} to a curve {{mvar|γ&#x303;}} in {{math|GL(''M'')}}.