Parallel Transport

The notion of parallel transport on a manifold  makes precise the idea of translating a vector field  along a differentiable curve to attain a new vector field  which is parallel to . More precisely, let  be a smooth manifold with affine connectionVector 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 covariantly constant, i.e.,  satisfies

(1)

for all  where, here,  denotes the unique covariant derivative of  associated to .

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 .

REFERENCES:

Do Carmo, M. Riemannian Geometry. Boston, MA: Birkhäuser, 1993.

Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd ed. Berkeley, CA: Publish or Perish Press, 1999.