Editing Parallel transport (section)

===Precise definition===

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

More precisely, if <math>\gamma:I\rightarrow M</math> is a [[curve|smooth curve]] parametrized by an interval <math>[a,b]</math> and <math>\xi\in T_xM</math>, where <math>x=\gamma(a)</math>, then a [[vector field]] <math>X</math> along <math>\gamma</math> (and in particular, the value of this vector field at <math>y=\gamma(b)</math>) is called the '''parallel transport of <math>\xi</math> along <math>\gamma</math>''' if
#<math>\nabla_{\gamma'(t)}X=0</math>, for all <math>t\in [a,b]</math>
#<math>X_{\gamma(a)}=\xi</math>.
Formally, the first condition means that <math>X</math> is parallel with respect to the [[pullback (differential geometry)|pullback connection]] on the [[pullback bundle]] <math>\gamma^* TM</math>. 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]]).

The parallel transport of <math>X \in T_{\gamma(s)} M</math> to the tangent space <math>T_{\gamma(t)} M</math> along the curve <math>\gamma : [0,1] \to M</math> is denoted by <math>\Gamma(\gamma)_s^t X</math>. The map
: <math>\Gamma(\gamma)_s^t : T_{\gamma(s)} M \to T_{\gamma(t)} M</math>
is linear. In fact, it is an isomorphism. Let <math>\overline\gamma : [0,1] \to M</math> be the inverse curve <math>\overline\gamma(t) = \gamma(1-t)</math>. Then <math>\Gamma(\overline\gamma)_t^s</math> is the inverse of <math>\Gamma(\gamma)_s^t</math>.

To summarize, 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'')}}.