Let $P$ be a principal $G$-bundle over $M$. Locally it looks like $M\times G$. Let $\pi$ and $\tilde{\pi}$ be the projections of $P$ onto $M$ and $G$, correspondingly. To my understanding, the connection form on $T$ can be defined as $$ \omega (X) =\tilde{\pi}^* \tilde{\omega}(X) = \tilde{\omega}(\tilde{\pi}_*X) \quad,\qquad X\in T_pP \quad, $$ where $\tilde{\omega}$ is the Maurer-Cartan form on $G$. From such a definition, it is clear that $\omega$ ignores the horizontal component $H_pP$ of the tangent space, and acts on the vertical component $V_pP$ as the Maurer-Cartan form.
- Double-check: is it true that $\omega(H_pP) = 0$?
Now, from (10.7) in Nakahara I learn that once we define a $\mathfrak{g}$-valued form $A$ on $M$, we can reconstruct $\omega$ as: $$ \omega=g^{-1}\pi^*Ag + g^{-1}\operatorname{d}g =g^{-1}A\pi_*g + g^{-1}\operatorname{d}g \quad. $$
Now, I would like to act with it on some vector from $T_pP$. Let $X=X_v+X_h\in T_pP$, where $X_v \in V_pP$ and $X_h\in H_pP$. I have then: $$ \omega(X_v+X_h) = g^{-1}A\pi_*(X_v)g + g^{-1}\operatorname{d}g(X_v) +g^{-1}A\pi_*(X_h)g + g^{-1}\operatorname{d}g(X_h) \quad. $$ The first term vanishes since $\pi_*X_v = 0$. The last term vanishes for $X_h$ has the zero component along the $V_pP$ (we have nothing to pull back). Second term is precisely what we need, the Maurer-Cartan form on $G$. My question is about the third term.
- What can you say about $g^{-1}A\pi_*(X_h)g$? Does it vanish? If so, how do we understand it? If not, how does it contribute to $\omega(X)$?