In my Global Analysis course there was the following exercice:
Show that the set of all connection on a bundle $E\to M$ is a non-trivial affine space based on $\Omega^1(\text{End}(E))$.
My teacher told us to use the fact that this set was convex, i.e. for $\nabla$, $\hat \nabla$, $t\nabla + (1 - t)\hat \nabla$ is also a connection for $t \in [0,1]$. However, I tried to do something and I do not use the convexity. This is what I've done:
Let $\psi_U$ be a local trivialization of $E$ over $U \subset M$. Then we can define a connection $\nabla_U$ on $E|_U$ by declaring $$\nabla_U s = \psi_U^{-1}(d(\psi_U(s)))$$ for $s \in \Gamma(E)$. Using a partition of unity, a collection of these local covariant derivatives can be sewed into a global covariant derivative. This show the non-emptyness. Then for $\nabla, \hat \nabla$, we have that $\nabla - \hat \nabla$ is a $C^\infty$-linear map and from there it is not too hard to prove the existence of an $a \in \Omega^1(\text{End}(E))$ such that $$\nabla - \hat \nabla = a.$$
Is my argument correct ? Do we need the convexity somewhere in the proof ?