Given a principal bundle 
, with fiber a Lie group 
and base manifold 
, and a group representation
of 
,
say 
,
then the associated vector bundle is
 |
(1)
|
In particular, it is the quotient space 
where 
.
This construction has many uses. For instance, any group representation of the orthogonal group gives rise to a bundle of tensors on a Riemannian manifold as the vector bundle associated to the frame bundle.
For example, 
is the frame bundle on 
, where
![pi([w_1; w_2; w_3])=w_1,](/images/equations/AssociatedVectorBundle/NumberedEquation2.svg) |
(2)
|
writing the special orthogonal matrix with rows 
. It is a 
bundle with the action defined by
![[costheta -sintheta; sintheta costheta]·A=[1 0 0; 0 costheta -sintheta; 0 sintheta costheta]A,](/images/equations/AssociatedVectorBundle/NumberedEquation3.svg) |
(3)
|
which preserves the map 
.
The tangent bundle is the associated vector bundle with the standard group representation of
on 
,
given by pairs 
,
with 
and 
.
Two pairs 
and 
represent the same tangent vector iff there is a 
such that 
and 
.
