I worked a little bit on the concepts of fiber bundles, associated bundles, tangent bundle, and connections. I studied that an ehresman connection on a principal bundle leads to the definition of a covariant derivative on an associated vector bundle. I get all the definition but in the whole reasoning there is a point where I am stuck.
It is when we define the Vertical and horizontal subspaces. In the 3 books on diff geometry I have, none provide a concrete example on how to determine the horizontal subspace. I understand that the Vertical one is defined canonically as $ker(\pi_*)$ but I have no clue how to provide one complement.
So I tried to work out a simple example but I get nowhere. I start with studying the bundle $\mathbb{R} \rightarrow \mathbb{R}$ with the identity projection map and the group $\{1\}$. I understand this is a trivial example but I end up finding that $Vert(\mathbb{R}) = 0$. So we have no choice of horizontal subspace.
1) I would like to get an example of fiber bundle where we are faced with the choice of horizontal subspace to define a connection.
2) I would also like to have the starting fiber bundle to study that would lead me to the definition of the covariant derivative being equal to the usual euclidean directional derivative. I am trying to apply the generalization to a concrete example.
Thanks