I'm reading Atiyah-Bott's paper "The Yang-Mills equations over Riemann surfaces" and have a couple of questions on page 547. They define a connection $A$ as a $G$-invariant splitting of the exact sequence
$ 0 \rightarrow T\_FP \rightarrow TP \rightarrow \pi^{-1}TM \rightarrow 0 $. 
I'm not sure how the group $G$ acts on $\pi^{-1}TM$. 


My guess is that if $h\_p = (p,\psi\_{\pi(p)})$ is in this space (where $p \in P$, and $\psi\_{\pi(p)} \in T\_{\pi(p)}M$) then $g \cdot h\_p = (pg, \psi\_{\pi(pg)=\pi(p)})$ . And I guess $G$-invariance means that if $X\_p \in T\_pP$ is uniquely written as $V\_p + H\_p$ (for $V\_p \in T\_FP\_p$ and $H\_p$ the image by $\gamma$, the right inverse of $\pi\_\*$, of an element in $\pi^{-1}TM$, say $h\_p$ ) then $G$-invariance means that $g v\_p$ is uniquely written as $gX\_{p} + g H\_p = X\_{pg} + \gamma(g h\_p) $.


On the same page they want to see the above exact sequence as a pull-back sequence of bundles over $M$. The middle term comes from a bundle $E(P)$ whose fibre over a point $\pi(p)$ should consist of the fibre $T\_pP$, but instead they say it is given as, $E(P)\_m = \Gamma \{ TP|\_{\pi^{-1}(m)} \}^G$. Why are they adding the $G$-invariant?



Definition of Connection as G-invariant splitting of a sequence which is a pulled back sequence of bundles
The bundle $\pi^{-1}TM$ is the set of tuples $(p,m,v)$ so that $m \in M$, $p \in \pi^{-1}\{m\}$, and $v \in T\_m M$. The $G$-action is $g(p,m,v)=(gp,m,v)$, only acting on $p$. So you are correct. They are correct (of course) that $E(P)\_m$ should consist of the $G$-equivariant sections of $TP$ along the fiber $P\_m = \pi^{-1}\{m\}$. In order to make bundles descend to $M$, not living on $P$, we have to work with $G$-equivariant objects along fibers over each point of $M$. That way we can work on $M$, not on $P$. (Not how I think about principal bundles and connections, but certainly something Atiyah always does.) 


