Various definitions of Connections on bundles

Question

Let $X$ be a smooth manifold and suppose I have a smooth vector bundle $E\to X$ which admits a connection $D$. Then on an open set $U\subset X$ where $E$ is trivial, once I choose a frame, say $e_1,...e_n$, I get a connection matrix by the rule $D(e_i)=\sum_{j=1}^{n}e_j\otimes\omega_{ji}^U$ (a column vector of 1 forms). If $V \subset X$ is another open set where $E$ is trivial and I choose a frame over $V$, say $f_1,...f_n$, then similarly I get $\omega^V$. Let $e_i=\sum_{j=1}^{n}g_{ji}f_j$. Then we have $\omega^V=g\omega^U g^{-1}-dgg^{-1}$.

On the other hand, we can form the associated principal $GL_n$ bundle $P\stackrel{\pi}{\to} X$, with $GL_n$ acting on the right. A connection on $P$ is a $GL_n$ invariant splitting of $\pi^*\Omega_X\to \Omega_P$. Locally, after we choose trivialisations for $P$ using the frames for $E$ chosen earlier, this becomes equivalent to giving a $GL_n$ equivariant splitting of $\Omega_U\to \Omega_U\oplus\Omega_G$, which is equivalent to giving a map $\Omega_{G,e}\to\Omega_{U,x}$ for each point $x\in U$, this gives the $\omega^U$. If $x\in U\cap V$, then the point $(x,A)$ in $U\times G$ is identified with the point $(x,gA)$ in $V\times G$. The gluing gives the following relation $\omega^V=g\omega^U g^{-1}+dgg^{-1}$. This is different from the one above. Is that OK?

Answer 1

I don't think there's any problem here. The question you want to ask is: Given a section $u$ of $E$, whether you get a consistent answer for $\nabla u$, using either frame. In other words, does $\nabla (u^ie_i) = \nabla (\hat{u}^i\hat{e}_i)$ using your change of frame formulas? I think you'll see what's going on if you work this out.

Let me elaborate a little bit. I never write change of frame formulas for the connection $1$-forms in isolation, because I am always confused by whether I should be acting on the frame by $g$ or $g^{-1}$. And whether $G$ is supposed to be acting on the right or left. But I know I will always get the right answer if I think about applying the connection to local section $u$ and expand it using two different local frames. Everything always automatically works out correctly when I do it this way.