# Affine space of connections

From Diffgeom

(Redirected from Affine space of all 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 and a vector bundle over , the **affine space of connections** on is defined as the set of all connections on , viewed as a subset of the vector space of all bilinear maps:

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*