Let be an smooth manifold with tangent bundle . For every we have the vector space , and we can consider the collection of all ordered basis of it. We will denote it . Observe that an ordered basis could be identified with a linear isomorphism (see note basis and change of basis).
Moreover, we have a natural action of into : for and we obtain . This action is free and transitive.
We define the frame bundle as the set
together with the natural projection.
What are the trivializations? Observe the picture:
We give the topology induced for the , and give to the final topology induced by the inclusions .