Frame bundle

The general linear group acts naturally on

via a change of basis, giving the frame bundle the structure of a principal

For this reason it is sometimes called the tangent frame bundle.

, has a natural right action by the general linear group

is both free and transitive (this follows from the standard linear algebra result that there is a unique invertible linear transformation sending one basis onto another).

although it lacks a group structure, since there is no "preferred frame".

This action is clearly free and the orbits are just the fibers of

can be given a natural topology and bundle structure determined by that of

is the final topology coinduced by the inclusion maps

as above, or more abstractly using the fiber bundle construction theorem.

is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as

One can easily check that this map is well-defined.

The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in

the coordinate vector fields define a smooth frame on

This relationship can be expressed by means of a vector-valued 1-form on

called the solder form (also known as the fundamental or tautological 1-form).

The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of

which are, in turn, in 1-1 correspondence with smooth bundle maps

As a naming convention, the term "tautological one-form" is usually reserved for the case where the form has a canonical definition, as it does here, while "solder form" is more appropriate for those cases where the form is not canonically defined.

is equipped with a Riemannian bundle metric then each fiber

It is then possible to talk about the set of all orthonormal frames for

is equipped with the standard Euclidean metric.

acts freely and transitively on the set of all orthonormal frames via right composition.

It can be constructed by a method entirely analogous to that of the ordinary frame bundle.

Again, the construction works just as well in the smooth category.

-dimensional Riemannian manifold, then the orthonormal frame bundle of

comes with additional structure it is often natural to consider a subbundle of the full frame bundle of

is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of

The orthonormal frame bundle is just a reduction of the structure group of

However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed.