Parallel Transport - Generalizations

Generalizations

The parallel transport can be defined in greater generality for other types of connections, not just those defined in a vector bundle. One generalization is for principal connections (Kobayashi & Nomizu 1996, Volume 1, Chapter II). Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω. As in the case of vector bundles, a principal connection ω on P defines, for each curve γ in M, a mapping

from the fibre over γ(s) to that over γ(t), which is an isomorphism of homogeneous spaces: i.e. for each g∈G.

Further generalizations of parallel transport are also possible. 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 though of as a map "rolling" a certain model space along a curve in the manifold. This rolling is called development.