Let's take the following definitions.
Definition 1. An Ehresmann connection is a smooth assignment of the complements of the vertical subspaces of the tangent spaces of the total space of the bundle. These complement subspaces are called vertical subspaces.
Definition 2. The horizontal lift of a differentiable curve $C$ in the base space is a curve $D$ in the total space having the following properties
- The tangent vectors of $D$ are horizontal at every point.
Definition 3. An Ehresmann connection on a smooth fiber bundle is curvature free if any horizontal lift of any closed differentiable curve in the base space is also closed. The connections which are not curvature free arre called curved.
Let $M$ be a simply connected 1-dimensional smooth manifold, and $\pi:E\to M$ a smooth fiber bundle.
In the special case when $E=\mathbb R^2$ , $M=\mathbb R$, and $\pi:(x,y)\mapsto (x,0)$, it is fairly obvious, that the horizontal lifts of every closed curve in $M$ are closed. But is this true also, when the fibers are not simply connected, or have more than 1-dimensions?