Theorem: Existence and uniqueness of horizontal lifts of curves

Let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$ , and let $H=\ker(\omega)\subset TP$ be its horizontal distribution.

Let $\gamma:[a,b]\to M$ be a smooth curve, and choose a point $p_0\in P$ with $\pi(p_0)=\gamma(a)$ .

There exists a unique smooth curve $\widetilde\gamma:[a,b]\to P$ such that:

$\pi\circ \widetilde\gamma=\gamma$ ,
$\widetilde\gamma(a)=p_0$ ,
$\dot{\widetilde\gamma}(t)\in H_{\widetilde\gamma(t)}$ for all $t$ (i.e. $\widetilde\gamma$ is horizontal).

Moreover, the lift depends smoothly on $(\gamma,p_0)$ under smooth variations.

Remarks

In a local trivialization $P|_U\cong U\times G$ , the horizontality condition becomes an ODE in $G$ driven by the local connection $1$ -form, so existence and uniqueness follow from standard ODE theory.
Horizontal lifting is the basic input for parallel transport on principal and associated bundles.

Examples

Trivial bundle with connection form. For $P=M\times G$ and a connection given by a $\mathfrak g$ -valued $1$ -form $A$ on $M$ , writing $\widetilde\gamma(t)=(\gamma(t),g(t))$ , horizontality is $g(t)^{-1}\dot g(t) = -A_{\gamma(t)}(\dot\gamma(t)),$ an ODE with unique solution given $g(a)$ .

Product (flat) connection. If $A=0$ , then the equation is $\dot g(t)=0$ , so the horizontal lift is simply $(\gamma(t),g_0)$ : constant group component.

Circle bundles. For a principal $U(1)$ -bundle with a connection $1$ -form, horizontal lifts of closed curves encode holonomy as a phase factor; this is the simplest instance of connection-induced transport.