Affine space of vector-bundle connections ($\mathrm{Conn}(E)={\nabla }^0+{\Omega }^1(M;\mathrm{End}E)$)

The difference of two connections on a vector bundle is a smooth one-form with values in the endomorphism bundle: the Leibniz derivative terms cancel, leaving pointwise linearity in both the tangent vector and section value. Conversely every such form added to a connection gives another connection. Thus the collection is an affine space with this model vector space. Choosing ${\nabla }^0$ supplies a noncanonical origin; there is no distinguished zero connection.