Parallel transport
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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.
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Parallel fields on smooth curves
A vector field <math>X</math> on a smooth curve <math>\gamma</math> is called parallel if
- <math>\nabla_{\dot\gamma(t)}X=0</math>
for any t.
Parallel transport
Let M be a smooth manifold with connection <math>\nabla</math>, <math>\gamma: I \to M</math> a smooth curve parameterized by the open interval I which includes 0 and let <math>X_0 \in \mathrm{T}_{\gamma(0)} M</math> be a tangent vector at γ(0). Then there exists a unique vector field X along <math>\gamma</math> such that <math>\nabla_{\dot{\gamma}} X = 0</math> and <math>X(0) = X_0</math>. The vector field X is called the parallel transport of <math>X_0</math> along <math>\gamma</math>.
Geodesics
Geodesics on (pseudo-)Riemannian manifolds are defined as follows. Let M be a smooth manifold with connection <math>\nabla</math>. A smooth curve <math>\gamma: I \longrightarrow M</math> is a geodesic if <math>\dot\gamma</math> (as a vector field along <math>\gamma</math>) is parallel along itself. In other words, if
- <math>\nabla_{\dot\gamma(t)}\dot\gamma = 0</math>
Parallel and geodesic vector fields
A vector field <math>X</math> on M is called parallel if
- <math>\nabla_Y X = 0</math> <math>\forall Y \in \mathrm{T}M</math>
and geodesic if
- <math>\nabla_X X = 0</math>.