Parallel transport
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.
Other notions of connection come equipped with their own parallel transportation systems as well.
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.
Given a fixed affine connection on a Riemannian manifold, there is a unique way to do parallel transport of tangent vectors.
[3] Different choices of affine connections will lead to different systems of parallel transport.
Intuitively speaking, parallel vector fields have all their derivatives equal to zero and are therefore in some sense constant.
However, in a local trivialization it is a first-order system of linear ordinary differential equations, which has a unique solution for any initial condition given by the second condition (for instance, by the Picard–Lindelöf theorem).
To summarize, parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve.
A linear isomorphism is determined by its action on an ordered basis or frame.
The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane
Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.
Since the first metric has zero curvature, the transport between two points along the circle could be accomplished along any other curve as well.
However, the second metric has non-zero curvature, and the circle is a geodesic, so that its field of tangent vectors is parallel.
, Taking the derivative at t = 0, the operator ∇ satisfies a product rule with respect to the metric, namely An affine connection distinguishes a class of curves called (affine) geodesics.
, that is Taking the derivative with respect to time, this takes the more familiar form If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.
More precisely, first note that if γ: I → M, where I is an open interval, is a geodesic, then the norm of
Let E → M be a vector bundle with connection ∇ and γ: I → M a smooth curve parameterized by an open interval I.
More precisely, X is the unique part of E along γ such that Note that in any given coordinate patch, (1) defines an ordinary differential equation, with the initial condition given by (2).
Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve: from the vector space lying over γ(s) to that over γ(t).
In particular, parallel transport around a closed curve starting at a point x defines an automorphism of the tangent space at x which is not necessarily trivial.
There is a close relation between this group and the value of the curvature of ∇ at x; this is the content of the Ambrose–Singer holonomy theorem.
Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition
Conversely, if a suitable notion of parallel transport is available, then a corresponding connection can be obtained by differentiation.
Consider an assignment to each curve γ in the manifold a collection of mappings such that The notion of smoothness in condition 3. is somewhat difficult to pin down (see the discussion below of parallel transport in fibre bundles).
In particular, modern authors such as Kobayashi and Nomizu generally view the parallel transport of the connection as coming from a connection in some other sense, where smoothness is more easily expressed.
One generalization is for principal connections (Kobayashi & Nomizu 1996, Volume 1, Chapter II).
In the context of Ehresmann connections, where the connection depends on a special notion of "horizontal lifting" of tangent spaces, one can define parallel transport via horizontal lifts.
Cartan connections are Ehresmann connections with additional structure which allows the parallel transport to be thought of as a map "rolling" a certain model space along a curve in the manifold.
