Connection (vector Bundle) - Affine Properties

Affine Properties

Every vector bundle admits a connection. However, connections are not unique. If ∇₁ and ∇₂ are two connections on E → M then their difference is a C∞-linear operator. That is,

for all smooth functions f on M and all smooth sections σ of E. It follows that the difference ∇₁ − ∇₂ is induced by a one-form on M with values in the endomorphism bundle End(E) = E⊗E*:

Conversely, if ∇ is a connection on E and A is a one-form on M with values in End(E), then ∇+A is a connection on E.

In other words, the space of connections on E is an affine space for Ω1(End E).