Frame bundle
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Mobius_frame_bundle.png/220px-Mobius_frame_bundle.png)
In
The frame bundle of a
Definition and construction
Let E → X be a real
The set of all frames at x, denoted Fx, has a natural
This action of GL(k, R) on Fx is both
The frame bundle of E, denoted by F(E) or FGL(E), is the disjoint union of all the Fx:
Each point in F(E) is a pair (x, p) where x is a point in X and p is a frame at x. There is a natural projection π : F(E) → X which sends (x, p) to x. The group GL(k, R) acts on F(E) on the right as above. This action is clearly free and the
The frame bundle F(E) can be given a natural topology and bundle structure determined by that of E. Let (Ui, φi) be a
given by
With these bijections, each π−1(Ui) can be given the topology of Ui × GL(k, R). The topology on F(E) is the final topology coinduced by the inclusion maps π−1(Ui) → F(E).
With all of the above data the frame bundle F(E) becomes a
The above all works in the smooth category as well: if E is a smooth vector bundle over a
Associated vector bundles
A vector bundle E and its frame bundle F(E) are associated bundles. Each one determines the other. The frame bundle F(E) can be constructed from E as above, or more abstractly using the fiber bundle construction theorem. With the latter method, F(E) is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as E but with abstract fiber GL(k, R), where the action of structure group GL(k, R) on the fiber GL(k, R) is that of left multiplication.
Given any
associated to F(E) which is given by product F(E) × V modulo the equivalence relation (pg, v) ~ (p, ρ(g)v) for all g in GL(k, R). Denote the equivalence classes by [p, v].
The vector bundle E is
where v is a vector in Rk and p : Rk → Ex is a frame at x. One can easily check that this map is
Any vector bundle associated to E can be given by the above construction. For example, the dual bundle of E is given by F(E) ×ρ* (Rk)* where ρ* is the dual of the fundamental representation. Tensor bundles of E can be constructed in a similar manner.
Tangent frame bundle
The tangent frame bundle (or simply the frame bundle) of a
Smooth frames
where p is a frame at x. It follows that a manifold is parallelizable if and only if the frame bundle of M admits a global section.
Since the tangent bundle of M is trivializable over coordinate neighborhoods of M so is the frame bundle. In fact, given any coordinate neighborhood U with coordinates (x1,…,xn) the coordinate vector fields
define a smooth frame on U. One of the advantages of working with frame bundles is that they allow one to work with frames other than coordinates frames; one can choose a frame adapted to the problem at hand. This is sometimes called the
Solder form
The frame bundle of a manifold M is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of M. This relationship can be expressed by means of a vector-valued 1-form on FM called the solder form (also known as the fundamental or tautological 1-form). Let x be a point of the manifold M and p a frame at x, so that
is a linear isomorphism of Rn with the tangent space of M at x. The solder form of FM is the Rn-valued 1-form θ defined by
where ξ is a tangent vector to FM at the point (x,p), and p−1 : TxM → Rn is the inverse of the frame map, and dπ is the
where Rg is right translation by g ∈ GL(n, R). A form with these properties is called a basic or
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. This convention is not being observed here.
Orthonormal frame bundle
If a vector bundle E is equipped with a
where Rk is equipped with the standard
The orthonormal frame bundle of E, denoted FO(E), is the set of all orthonormal frames at each point x in the base space X. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank k Riemannian vector bundle E → X is a principal O(k)-bundle over X. Again, the construction works just as well in the smooth category.
If the vector bundle E is orientable then one can define the oriented orthonormal frame bundle of E, denoted FSO(E), as the principal SO(k)-bundle of all positively oriented orthonormal frames.
If M is an n-dimensional Riemannian manifold, then the orthonormal frame bundle of M, denoted FOM or O(M), is the orthonormal frame bundle associated to the tangent bundle of M (which is equipped with a Riemannian metric by definition). If M is orientable, then one also has the oriented orthonormal frame bundle FSOM.
Given a Riemannian vector bundle E, the orthonormal frame bundle is a principal O(k)-subbundle of the general linear frame bundle. In other words, the inclusion map
is principal
G-structures
If a smooth manifold M comes with additional structure it is often natural to consider a subbundle of the full frame bundle of M which is adapted to the given structure. For example, if M is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of M. The orthonormal frame bundle is just a reduction of the structure group of FGL(M) to the orthogonal group O(n).
In general, if M is a smooth n-manifold and G is a
over M.
In this language, a Riemannian metric on M gives rise to an O(n)-structure on M. The following are some other examples.
- Every oriented manifold has an oriented frame bundle which is just a GL+(n, R)-structure on M.
- A volume form on M determines a SL(n, R)-structure on M.
- A 2n-dimensional symplectic manifold has a natural Sp(2n, R)-structure.
- A 2n-dimensional complex or almost complex manifold has a natural GL(n, C)-structure.
In many of these instances, a G-structure on M uniquely determines the corresponding structure on M. For example, a SL(n, R)-structure on M determines a volume form on M. However, in some cases, such as for symplectic and complex manifolds, an added
References
- Kobayashi, Shoshichi; Nomizu, Katsumi (1996), ISBN 0-471-15733-3
- Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2008-08-02
- ISBN 0-8218-1385-4