Associated Vector Bundle
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المرجع الالكتروني للمعلوماتيه
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www.almerja.com
الجزء والصفحة:
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22-5-2021
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Associated Vector Bundle
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,](https://mathworld.wolfram.com/images/equations/AssociatedVectorBundle/NumberedEquation2.gif) |
(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,](https://mathworld.wolfram.com/images/equations/AssociatedVectorBundle/NumberedEquation3.gif) |
(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 
.
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