Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.


If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma\_{ij}^k (p) = 0 \ , $$ where $\Gamma\_{ij}^k$ denote the Christoffel symbols on those coordinates.


I am wondering whether a similar statement holds for principal connections. 


To be precise, let $G$ be a Lie group and let $P\_0 := G \times X \to X$ be the trivial principal $G$-bundle. Consider a principal connection on it, defined by a 1-form $\alpha$ on $P\_0$ with values on the Lie algebra $\mathfrak{g}$ of $G$. 


If $(g,x)\in P\_0$ is any point, is it possible to find a connection $\bar{\alpha}$ isomorphic to $\alpha$, whose value $\bar{\alpha}\_{(g,x)} $ at that point is zero?



Normalizing the value of a principal connection at a point
Yes,
consider a chart centered at $p$ and the rays emitting from $p$ with respect to this chart. Now, take a frame at $p$ and consider the parallel transport along
the rays. This gives you a local section of $P$ (or a new trivialisation if you prefer), and the connection 1-form with respect to this section vanishes at $p$.


