Connections on principal bundles: Local and Global Formulations. The standard definition of a connection on a principal $G$-bundle $\pi : P \to X$ is a smooth family of subspaces $H_{p}$ of $T_p P$ such that for every $p \in X$ we have a splitting of vector spaces $T_p P = H_p \oplus \mathrm{ker} (\pi_{*})_{p}$ and so that $H_{p g} = (R_{g})_{*} H_{p}$, where $R_g$ is the right action of $G$ on $P$. Equivalently, this is the same as specifying, for every $p \in X$, a right inverse $q_p : T_{\pi(p)} X \to T_p P$ for $(\pi_{*})_p$, or a left inverse $\theta_{p}: T_p P \to \mathrm{ker} (\pi_{*})_{p}$ for the inclusion, by standard theory for exact sequences of vector spaces.
All of the above works `fibrewisely'; I would like to know what the global picture is:
When will the pullback bundle $\pi^{*}(TX)$ and the kernel $T_V P$ of $\pi_{*}$ be subbundles of $TP$, so that we may write $TP  = T_{V}P \oplus \pi^{*}(TX) $, where this splitting is $G$-invariant? Will this only occur if we can find a $G$-invariant right bundle-morphism inverse for $\pi_{*}$, or a $G$-invariant left bundle-morphism inverse for the inclusion of $T_V P$?
More generally, the transition between the local and fibrewise pictures in the category of vector bundles over smooth manifolds is unclear to me, in particular the theory for short exact sequences - is there some standard reference that explains this in detail?
 A: The problem is that $\pi^*(TX)$ is not a subbundle of $TP$. There is a natural vector bundle homomorphism $TP\to\pi^*(TX)$, which is just an encoding of the tangent map $T\pi$. (By definition, the fiber of $\pi^*(TX)$ in a point $u\in P$ is $T_{\pi(u)}X$, so one can view $T_u\pi$ as a map between the fibers of $TP$ and $\pi^*(TX)$.) The kernel of this vector bundle homomorphism by definition is the vertical subbundle $T_VP\subset TP$ (which canonically is a subbundle of $TP$). This is usually expressed by the fact that 
$$
0\to T_VP\to TP\to\pi^*(TX)\to 0
$$
is a short exact sequence of vector bundles over $P$. This just meanst that the arrows are vector bundle homomorphisms (over the identity map on $P$), the one $T_VP\to TP$ is fiber-wise injective, its image in each fiber coincides with the kernel of the morphism $TP\to\pi^*(TX)$, and the latter morphism is fiber-wise surjective. (So the whole "exact sequence of vector bundles"-stuff is just a neat way to express the fiber-wise statements.) 
To define a connection (in the general fiber bundle sense), you just need a splitting of the above sequence. As in the case of vector spaces, this can be equivalently expressed as a left inverse to the map $T_VP\to TP$ ("vertical projection") or as a right inverse to $TP\to\pi^*(TX)$ ("specifying a horizontal subspace"). As before, this just encodes fiber-wise statements. For a principal connection, you in addition require compatibility with the principal right action of the structure group. 
The terminology "will this occur only if we can find ..." that you use seems a bit misleading to be. There always exist principal connections, but there are many of them, so usually this is a choice.  
A: Atiyah sequence is the notion that you might be heading to.
Take a look at wikipedia, actually in the references cited there.
https://en.wikipedia.org/wiki/Atiyah_algebroid
This question at Mathoverflow has a nice answer by Florian, although people there did not see the relationship of it and the Atiyah sequence, this might be the reason why he gets downvotes. 
https://mathoverflow.net/questions/58581/existence-of-connections-on-principal-bundles 
