In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay "parallel" with respect to the given connection. A field <math>V<math> on a smooth curve <math>\gamma<math> is called parallel if

<math>\nabla_{\dot\gamma(t)}V=0<math>

for any t.

See also: Connection (mathematics), Covariant derivative