Let $E$ be a rank-$k$ vector bundle over a smooth manifold $M$, and let $∇$ be a connection on $E$. Given a piecewise smooth loop (topology) $γ : [0,1] → M$ based at $x$ in $M$, the connection defines a parallel transport map $P_γ : E_x → E_x$. This map is both linear and invertible, and so defines an element of the general linear group $GL(E_x)$. The holonomy group of $∇$ based at $x$ is defined as $$ \operatorname{Hol}_x(\nabla) = \{P_\gamma \in \mathrm{GL}(E_x) \mid \gamma \text{ is a loop based at } x\}. $$
My question is that:
the holonomy of the connection can be identified with a Lie group, the holonomy group. Why is this true? (Could we write an explicitly Lie algebra around the point $x$?) What will be the global structure of the Lie group (beyond the local Lie algebra?)
Here $E$ be a rank-$k$ vector bundle. Do we have the holonomy group as a Lie group only for this case? Or do we have other connection $∇$ on any other $E$, to give that $\operatorname{Hol}_x(\nabla)$ is still a Lie group? Namely, for what constraints on $E$?
The Lie algebra of the honomy group at a point is described by the curvature of the connectiion see Ambrose-Singer.
https://en.wikipedia.org/wiki/Holonomy#Ambrose%E2%80%93Singer_theorem