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)
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)