Editing Parallel transport (section)

==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 [[connection (principal bundle)|principal connections]] {{harv|Kobayashi|Nomizu|1996|loc=Volume 1, Chapter II}}. Let ''P'' &rarr; ''M'' be a [[principal bundle]] over a manifold ''M'' with structure [[Lie group]] ''G'' and a principal connection &omega;. As in the case of vector bundles, a principal connection &omega; on ''P'' defines, for each curve &gamma; in ''M'', a mapping
:<math>\Gamma(\gamma)_s^t : P_{\gamma(s)} \rightarrow P_{\gamma(t)}</math>
from the fibre over &gamma;(''s'') to that over &gamma;(''t''), which is an isomorphism of [[homogeneous space]]s: i.e. <math>\Gamma_{\gamma(s)} gu = g\Gamma_{\gamma(s)}</math> for each ''g''&isin;''G''.

Further generalizations of parallel transport are also possible. In the context of [[Ehresmann connection]]s, where the connection depends on a special notion of "[[horizontal space|horizontal lift]]ing" of tangent spaces, one can define [[Ehresmann connection#Parallel transport via horizontal lifts|parallel transport via horizontal lifts]]. [[Cartan connection]]s are Ehresmann connections with additional structure which allows the parallel transport to be thought of as a map "rolling" a certain [[Klein geometry|model space]] along a curve in the manifold. This rolling is called [[development (differential geometry)|development]].