Affine space of connections

This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

Definition

Given a differential manifold $M$ and a vector bundle $E$ over $M$, the affine space of connections on $M$ is defined as the set of all connections on $M$, viewed as a subset of the vector space of all bilinear maps:

$\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$

This subset is in fact an affine space, because the condition of being a difference of two connections is a linear system of conditions. Fill this in later