Parallel transport

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In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with a covariant derivative (in the tangent bundle), then the connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection. Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.

The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one -- one way of connecting up the geometries of points on a curve -- is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, vice versa, parallel transport is the local realization of a connection.

As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose-Singer theorem makes explicit this relationship between curvature and holonomy.

Contents

1 Parallel transport for the covariant derivative

1.1 Geodesics
1.2 Parallel and geodesic vector fields

2 See also

Parallel transport for the covariant derivative

Let M be a smooth manifold with covariant derivative ∇ and γ : I → M a smooth curve parameterized by the open interval I. A vector field along γ is called parallel if

[\nabla_X=0], for all t.

Suppose we are given a tangent vector X₀ ∈ T_PM at P = γ(0) ∈ M, rather than a vector field. The parallel transport of X₀ along γ is the extension of X₀ to a parallel vector field X on γ such that X_γ(0) = X₀.

Stated another way, X is the unique vector field along γ such that

[\nabla_} X = 0] (1), and

[X(0) = X_0]. (2)

Note that in local coordinates (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–Lindelöf theorem guarantees the existence and uniqueness of the solution.

Geodesics

Geodesics on (pseudo-)Riemannian manifolds are defined as follows. Let M be a smooth manifold with connection [\nabla]. A smooth curve [\gamma: I \longrightarrow M] is a geodesic if [\dot\gamma] (as a vector field along [\gamma]) is parallel along itself. In other words, if

[\nabla_\dot\gamma = 0]

Parallel and geodesic vector fields

A vector field [X] on M is called parallel if

[\nabla_Y X = 0] [\forall Y \in \mathrmM]

and geodesic if

[\nabla_X X = 0].

Parallel transport

Parallel transport for the covariant derivative

Geodesics

Parallel and geodesic vector fields

See also