The notion of parallel transport on a manifold makes precise the idea of translating
a vector field along a differentiablecurve to attain a new vector field which is parallel to
.
More precisely, let
be a smooth manifold with affineconnectionVector Bundle Connection , let be a differentiable
curve from an interval into , and let be a vector
tangent to
at
for some .
A vector field
is said to be the parallel transport of along provided that , , is a vector field for which .

Note that the use of the quantifier parallel in the above definition makes reference to the fact that a parallel transport of a vector field along a curve is necessarily covariantlyconstant, i.e., satisfies

A standard result in differential geometry
is that, under the above hypotheses, parallel transports are unique.

In addition to the above definition, some literature defines parallel transport in a more function analytic way. Indeed, given an interval and a point , a parallel transport of along is nothing more than a linear transformation

(2)

which maps
to .
It is obvious that this transformation is invertible, its inverse being given simply
by parallel transport along the reversed portion of from to . The expression has added benefit, too, because despite being defined
intrinsically in terms of the affine connection on , it also provides a mechanism whereby one can recover
a manifold's affine connection given a collection of parallel vector fields along a curve . In particular, if and , then

(3)

where
is the desired vector field given by the connection and where .