Editing Connection (vector bundle) (section)

==Relation to principal and Ehresmann connections==

Let <math>E\to M</math> be a vector bundle of rank <math>k</math> and let <math>\mathcal{F}(E)</math> be the [[frame bundle]] of <math>E</math>. Then a [[connection (principal bundle)|(principal) connection]] on <math>\mathcal{F}(E)</math> induces a connection on <math>E</math>. First note that sections of <math>E</math> are in one-to-one correspondence with [[equivariant|right-equivariant]] maps <math>\mathcal{F}(E)\to \mathbb{R}^k</math>. (This can be seen by considering the [[pullback bundle|pullback]] of <math>E</math> over <math>\mathcal{F}(E)\to M</math>, which is isomorphic to the [[trivial bundle]] <math>\mathcal{F}(E)\times  \mathbb{R}^k</math>.) Given a section <math>s</math> of <math>E</math> let the corresponding equivariant map be <math>\psi(s)</math>. The covariant derivative on <math>E</math> is then given by
:<math>\psi(\nabla_Xs) = X^H(\psi(s))</math>
where <math>X^H</math> is the [[vertical and horizontal bundles|horizontal lift]] of <math>X</math> from <math>M</math> to <math>\mathcal{F}(E)</math>. (Recall that the horizontal lift is determined by the connection on <math>\mathcal{F}(E)</math>.)

Conversely, a connection on <math>E</math> determines a connection on <math>\mathcal{F}(E)</math>, and these two constructions are mutually inverse.

A connection on <math>E</math> is also determined equivalently by a [[Ehresmann connection#Vector bundles and covariant derivatives|linear Ehresmann connection]] on <math>E</math>. This provides one method to construct the associated principal connection.

The induced connections discussed in [[#Induced connections]] can be constructed as connections on other associated bundles to the frame bundle of <math>E</math>, using representations other than the standard representation used above. For example if <math>\rho</math> denotes the standard representation of <math>\operatorname{GL}(k,\mathbb{R})</math> on <math>\mathbb{R}^k</math>, then the associated bundle to the representation <math>\rho \oplus \rho</math> of <math>\operatorname{GL}(k,\mathbb{R})</math> on <math>\mathbb{R}^k \oplus \mathbb{R}^k</math> is the direct sum bundle <math>E\oplus E</math>, and the induced connection is precisely that which was described above.