The frame bundle

Let M be an smooth manifold with tangent bundle TM. For every pM we have the vector space TpM, and we can consider the collection of all ordered basis of it. We will denote it FpM. Observe that an ordered basis could be identified with a linear isomorphism b:RnTpM (see note basis and change of basis).

Moreover, we have a natural action of GL(n,R) into FpM: for f:RnTpM and TGL(n,R) we obtain fT:RnTpM. This action is free and transitive.

We define the frame bundle FM as the set

FM=pMFpM

together with the natural projection.

What are the trivializations? Observe the picture:
framebundle.jpg

We give π1(Ui) the topology induced for the {Ψi}, and give to FM the final topology induced by the inclusions π1(Ui)FM.

The frame bundle is an example of principal bundle.

Tautological 1-form

The frame bundle has a solder form that is natural. It is explained here