# Normalizing the value of a principal connection at a point

Let $P_0 := G \times X \to X$ be the trivial principal $G$-bundle, and 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 always possible to find a connection $\bar{\alpha}$ isomorphic to $\alpha$, whose value $\bar{\alpha}_{(g,x)}$ at that point is zero?

(Two connections $\alpha, \bar{\alpha}$ on $P_0$ are isomorphic if there exists an isomorphism of principal bundles $\phi \colon P_0 \to P_0$ such that $\phi^* (\alpha ) = \bar{\alpha}$).

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