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# 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 .

## See also

Covariant Derivative, Manifold, Parallel Vectors, Vector field

This entry contributed by Christopher Stover

## 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.

## Cite this as:

Stover, Christopher. "Parallel Transport." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ParallelTransport.html