Frame bundle

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a frame bundle is a <a href="/facts/Principal_fiber_bundle/bbB2D94B">principal fiber bundle</a> 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 associated with any <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a> 
 
 
 
 E
 
 
 {\displaystyle E}
 
. The fiber of 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 over a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is the set of all <a href="/facts/Ordered_basis/89IPoN6c">ordered bases</a>, or frames, for 
 
 
 
 
 E
 
 x
 
 
 
 
 {\displaystyle E_{x}}
 
. The <a href="/facts/General_linear_group/B54gNILS">general linear group</a> acts naturally on 
 
 
 
 F
 (
 E
 )
 
 
 {\displaystyle F(E)}
 
 via a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a>, giving the frame bundle the structure of a principal 
 
 
 
 
 G
 L
 
 (
 k
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (k,\mathbb {R} )}
 
-bundle (where k is the rank of 
 
 
 
 E
 
 
 {\displaystyle E}
 
).
The frame bundle of a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> is the one associated with its <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a>. For this reason it is sometimes called the tangent frame bundle.

Frame bundle open-in-new

Frame bundle