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).
Read more about this topic: Connection (vector Bundle)
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)