Let $M$ be a connected smooth paracompact manifold, $E$ a vector bundle over $M$ with fibre $\mathbb R^k$, and $\nabla$ a connection on $E$. It is known that Hol$^0(\nabla)$ is a connected Lie subgroup of $GL(k,\mathbb R)$. How can we show Hol$^0(\nabla)$ is an identity component of Hol$(\nabla)$?
It seems to me that there are two ways to understand this. The first way is to regard Hol$^0(\nabla)$ and Hol$(\nabla)$ as topological subspaces of $GL(k,\mathbb R)$. Another way is to make Hol$(\nabla)$ a Lie subgroup of $GL(k,\mathbb R)$ by left translating the differential structure of Hol$^0(\nabla)$. But to prove Hol$(\nabla)$ is a Lie group, one has to prove for any $a\in$ Hol$(\nabla)$, the mapping from Hol$^0(\nabla)$ to Hol$^0(\nabla)$ defined by $x \rightarrow axa^{-1}$ is differentiable. I am stuck here.