# Normalizing the value of a principal connection at a point

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?

• A principal connection form $\alpha$ maps isomorphically every fibre of the vertical tangent bundle $VP_{0} \cong P_{0}\times \mathfrak{g}$ to the Lie algebra $\mathfrak{g}$, so $\alpha$ cannot vanish at any point. – Oldřich Spáčil Jan 27 '14 at 12:01

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 $p$.
• I guess by $p$ you mean a point on the base $X$ and you want to say that the pullback of the connection form $\alpha$ along your local section vanishes at $p$? – Oldřich Spáčil Jan 27 '14 at 11:57
• To find such a section, you don't need to do anything with parallel transport (which involves solving differential equations). It suffices to take any section $\sigma:X\to P_0$ such that the tangent space of the image $\sigma(X)\subset P_0$ at $(g,x)$ is equal to the kernel of $\alpha_{(g,x)}:T_{(g,x)}\to{\frak{g}}$, for then $\sigma^*(\alpha)$ will vanish at $x$, which is what you really want (instead of $\alpha_{(g,x)}$ vanishing, which, as Oldfich noted in his comment above, never happens). – Robert Bryant Jan 27 '14 at 12:24
• You are all right: I'm not used to this language yet and, by trying to simplify the question, I got confused. What I was trying to argue was the existence of a section $\sigma \colon X \to P_0$ such that $\alpha_{|\sigma}$ vanishes at $p$, and any of your reasonings is enough. – José Navarro Jan 27 '14 at 16:25