Frame and coframe bundles
Given a -vector bundle of finite rank , its frame bundle (or bundle of frames in ) is the bundle over the same base whose fiber over is the set of all vector space bases of . The frame bundle has a natural action of given by an ordered change of basis which is free and transitive, i. e., the frame bundle is a principal -bundle.
The frame bundle of a manifold is the principal bundle (also denoted ) of frames in the tangent bundle .
In the finite-dimensional case, the dual -principal bundle is the coframe bundle of the manifold. This means that is the associated bundle to where the left action of on is given by right multiplication by inverses . Also using the same formula. Furthermore, the right action of on this associated bundle is given by left multiplication by inverses on factor.
Coframe bundle has the following independent description. One looks at the set of tuples of the form where and is chart of the smooth structure on , , (an atlas where -s make a basis of topology suffices). acts on the right on by
Then holds. The total space of the coframe bundle by the definition, as a set, consists of classes of equivalence of tuples in where iff and the Jacobian matrix of the transition between charts at is the unit matrix: . The left action of is induced on the quotient. There is an obvious projection . To define the differential and principal bundle structure one charts with local trivializations from the neighborhoods of the form , transfers the structure and checks that the transition functions are of the appropriate smoothness class and right -equivariant. The basic prescription is that to every chart one defines a map
where with . This does not depend on the choice of the chart around . There is an equivariance
and on intersection of and
Then is onto and
what shows that the transition functions are smooth (where has the standard differential structure).
In differential cohesion
Formalization of frame bundles in differential cohesion is discussed there in the section Differential cohesion – Frame bundles
The canonical differential 1-form
The frame bundle carries a canonical differential 1-form with values in .
This is defined as follows. Let be a point in the frame bundle over some point , hence a linear isomorphism . For a tangent vector to the frame bundle, its projection is a tangent vector to . Then the value of on is the image of this under the isomorphism
(Sternberg 64, section VII, (2.2))
Relation to -structures
A choice sub-bundle of a frame bundle which is a -principal bundle for defines a G-structure. See there for more.