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Diffgeom β

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:

\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