Parallel transport on a manifold 
makes precise the idea of translating a vector
field 
along a smooth curve to attain a new vector
field 
which is parallel to 
.
More precisely, let 
be a smooth manifold with affine connection 
, let 
be a smooth curve
from an interval 
into 
, and let 
be a vector
tangent to 
at 
for some 
.
A vector field 
along 
is the parallel transport of 
provided that 
and 
is covariantly constant, meaning that
 |
(1)
|
for all 
, where 
denotes the covariant
derivative of 
along 
associated to 
.
A standard result in differential geometry is that, under the above hypotheses, parallel transports are unique.
In addition to the transported vector field, parallel transport can be viewed in a functional analytic way. Given 
, parallel transport along 
from 
to 
is the linear transformation
 |
(2)
|
which maps 
to 
.
It is obvious that this linear transformation
is invertible, its inverse
being given simply by parallel transport along the reversed portion of 
from 
to 
. The expression 
also provides a way to recover the affine connection
from its parallel transport along all curves. In particular,
choose the reference parameter 
, let 
and 
, and let 
be a smooth vector field.
Then
 |
(3)
|
where 
is the desired tangent vector at 
given by the connection 
and where 
.
