Paralleltransport

In the differential geometry of parallel transport refers to a method of transporting geometrical objects along smooth curves in a manifold. If the manifold has a covariant derivative ( in the tangent bundle ), then you can vectors in the manifold along curves so transported that they stay parallel with respect to the members of the covariant derivative context.

Accordingly, one can construct for each context a parallel transport. A Cartan connection even allows the lifts of curves from the manifold into the associated principal bundle. Such Kurvenliftung allows the parallel transfer of reference frames, that is, the transport of a base of one point to another.

The belonging to a context parallel transport allows, in a way to move the local geometry of a manifold along a curve.

Just like a parallel transport can be constructed from a context that can be reversed from a parallel transport a related construct. In this respect, a relationship is an infinitesimal analogue of a parallel transport or parallel transport, the local realization of a relationship.

Besides the local realization of a connection, a parallel transport also provides a local realization of the curvature, the holonomy. The Ambrose -Singer theorem makes explicit the relationship between curvature and holonomy.

Parallel vector field

Let be a differentiable manifold with connection. A vector field along a curve is called parallel along, if for all. A field vector is parallel, if it is parallel with respect to each curve in the manifold.

Parallel transport

Let be a differentiable manifold, a curve and two real numbers. Then there exists a unique parallel to each vector field along so true. With the help of this existence and uniqueness result can be given that is called parallel transport to define. The figure

Which a vector of its unique parallel vector field evaluated at the point assigns.

The parallel transport for the Levi- Civita connection

The most important special case of the parallel transport is the transport of a tangent vector along a curve on a Riemannian manifold, the connection is the Levi- Civita connection.

Specifically: If a tangent vector at the point and a smooth curve, so called a vector field along, ie with, if and parallel transport of if:

So if the covariant derivative of along disappears.

This is a linear first order initial value problem, from which one can show the existence and uniqueness of a solution.

The magnitude of a vector, which is shifted in parallel is constant:

The parallel transport along a geodesic

In the event that a geodesic is the parallel transport has special properties.

For example, the tangent vector of a parameterized proportional to arc-length geodesic is itself parallel:

For this was precisely the definition of a geodesic on a Riemannian manifold.

The angle between the tangent vector of the geodesic and the vector is constant, since the amounts of both vectors are also constant ( see above).

Parallel transport in Euclidean spaces

In Euclidean space, the covariant derivative of the normal discharge in a particular direction. It disappears when, apart from the base point is constant, ie if all vectors are parallel.

The parallel transfer is thus a generalization of the parallel displacement of a vector along a curve.