# Affine space of connections

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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:

$\Gamma(TM) \times \Gamma(E) \to \Gamma(E)$

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