Parallel transport for an Ehresmann connection

Let $\pi:E\to M$ be a surjective submersion equipped with an Ehresmann connection. For a smooth curve $\gamma:[0,1]\to M$ and an initial point $e_0\in E_{\gamma(0)}$ , let $\widetilde\gamma$ denote the horizontal lift of the curve with $\widetilde\gamma(0)=e_0$ , defined (at least) on $[0,1]$ whenever the lift exists globally.

Let $D_\gamma\subset E_{\gamma(0)}$ be the open set of initial points whose horizontal lifts exist throughout $[0,1]$ . Definition. The parallel transport along $\gamma$ is the map

P_\gamma: D_\gamma \longrightarrow E_{\gamma(1)},\qquad P_\gamma(e_0):=\widetilde\gamma(1),

where $\widetilde\gamma$ is the horizontal lift starting at $e_0$ .

Invertibility and completeness

For the reversed curve $\bar\gamma(t)=\gamma(1-t)$ , uniqueness of horizontal lifts gives a diffeomorphism $P_\gamma:D_\gamma\to D_{\bar\gamma}$ with inverse $P_{\bar\gamma}$ . It is a diffeomorphism between the entire fibers when lifts exist throughout $[0,1]$ for every initial point in both directions. This holds for a complete connection. Forward existence alone need not give surjectivity. If the connection comes from a vector bundle connection, $P_\gamma$ is linear on each fiber; if it comes from a principal connection, it is $G$ -equivariant.

Examples

Product bundle. For $E=M\times F$ with the product connection, $P_\gamma$ is the identity map on the fiber $F$ : it sends $(\gamma(0),f)$ to $(\gamma(1),f)$ .
Levi-Civita transport on tangent vectors. For the Levi-Civita connection on the tangent bundle, $P_\gamma$ transports tangent vectors along $\gamma$ by keeping them covariantly constant; on a sphere this produces the classical “rotation” effect around a loop.
Line bundle with a local 1-form. In a trivialization along $\gamma$ where a complex line-bundle connection is $\nabla=d+A$ , parallel transport multiplies the fiber coordinate by $\exp(-\int_\gamma A)$ . For a Hermitian connection in a unitary frame, $A$ is imaginary-valued and this multiplier is a phase. A general complex connection can also change its magnitude.

Remarks

For a complete Ehresmann connection, transports around loops based at $x$ form a subgroup of the diffeomorphism group of the fiber $E_x$ . In the principal-bundle case this gives the holonomy subgroup of the structure group after choosing a fiber point. Curvature measures local dependence on contractible loops; even a flat connection can have nontrivial monodromy around noncontractible loops.