# Frame bundle

In mathematics, a **frame bundle** is a principal fiber bundle F(*E*) associated to any vector bundle *E*. The fiber of F(*E*) over a point *x* is the set of all ordered bases, or *frames*, for *E*_{x}. The general linear group acts naturally on F(*E*) via a change of basis, giving the frame bundle the structure of a principal GL(*k*, **R**)-bundle (where *k* is the rank of *E*).

The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the **tangent frame bundle**.

## Definition and construction

Let *E* → *X* be a real vector bundle of rank *k* over a topological space *X*. A **frame** at a point *x* ∈ *X* is an ordered basis for the vector space *E*_{x}. Equivalently, a frame can be viewed as a linear isomorphism

The set of all frames at *x*, denoted *F*_{x}, has a natural right action by the general linear group GL(*k*, **R**) of invertible *k* × *k* matrices: a group element *g* ∈ GL(*k*, **R**) acts on the frame *p* via composition to give a new frame

This action of GL(*k*, **R**) on *F*_{x} 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). As a topological space, *F*_{x} is homeomorphic to GL(*k*, **R**) although it lacks a group structure, since there is no "preferred frame". The space *F*_{x} is said to be a GL(*k*, **R**)-torsor.

The **frame bundle** of *E*, denoted by F(*E*) or F_{GL}(*E*), is the disjoint union of all the *F*_{x}:

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 orbits are just the fibers of π.

The frame bundle F(*E*) can be given a natural topology and bundle structure determined by that of *E*. Let (*U*_{i}, φ_{i}) be a local trivialization of *E*. Then for each *x* ∈ *U*_{i} one has a linear isomorphism φ_{i,x} : *E*_{x} → **R**^{k}. This data determines a bijection

given by

With these bijections, each π^{−1}(*U*_{i}) can be given the topology of *U*_{i} × GL(*k*, **R**). The topology on F(*E*) is the final topology coinduced by the inclusion maps π^{−1}(*U*_{i}) → F(*E*).

With all of the above data the frame bundle F(*E*) becomes a principal fiber bundle over *X* with structure group GL(*k*, **R**) and local trivializations ({*U*_{i}}, {ψ_{i}}). One can check that the transition functions of F(*E*) are the same as those of *E*.

The above all works in the smooth category as well: if *E* is a smooth vector bundle over a smooth manifold *M* then the frame bundle of *E* can be given the structure of a smooth principal bundle over *M*.

## 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 linear representation ρ : GL(*k*, **R**) → GL(*V*,**F**) there is a vector bundle

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 naturally isomorphic to the bundle F(*E*) ×_{ρ} **R**^{k} where ρ is the fundamental representation of GL(*k*, **R**) on **R**^{k}. The isomorphism is given by

where *v* is a vector in **R**^{k} and *p* : **R**^{k} → *E*_{x} is a frame at *x*. One can easily check that this map is well-defined.

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*} (**R**^{k})* 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 manifold *M* is the frame bundle associated to the tangent bundle of *M*. The frame bundle of *M* is often denoted F*M* or GL(*M*) rather than F(*TM*). If *M* is *n*-dimensional then the tangent bundle has rank *n*, so the frame bundle of *M* is a principal GL(*n*, **R**) bundle over *M*.

### Smooth frames

Local sections of the frame bundle of *M* are called smooth frames on *M*. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in *U* in *M* which admits a smooth frame. Given a smooth frame *s* : *U* → F*U*, the trivialization ψ : F*U* → *U* × GL(*n*, **R**) is given by

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 (*x*^{1},…,*x*^{n}) 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 method of moving frames.

### 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 F*M* 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 **R**^{n} with the tangent space of *M* at *x*. The solder form of F*M* is the **R**^{n}-valued 1-form θ defined by

where ξ is a tangent vector to F*M* at the point (*x*,*p*), *p*^{−1} : T_{x}*M* → **R**^{n} is the inverse of the frame map, and dπ is the differential of the projection map π : F*M* → *M*. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of π and right equivariant in the sense that

where *R*_{g} is right translation by *g* ∈ GL(*n*, **R**). A form with these properties is called a basic or tensorial form on F*M*. Such forms are in 1-1 correspondence with *TM*-valued 1-forms on *M* which are, in turn, in 1-1 correspondence with smooth bundle maps *TM* → *TM* over *M*. Viewed in this light θ is just the identity map on *TM*.

## Orthonormal frame bundle

If a vector bundle *E* is equipped with a Riemannian bundle metric then each fiber *E*_{x} is not only a vector space but an inner product space. It is then possible to talk about the set of all of orthonormal frames for *E*_{x}. An orthonormal frame for *E*_{x} is an ordered orthonormal basis for *E*_{x}, or, equivalently, a linear isometry

where **R**^{k} is equipped with the standard Euclidean metric. The orthogonal group O(*k*) acts freely and transitively on the set of all orthonormal frames via right composition. In other words, the set of all orthonormal frames is a right O(*k*)-torsor.

The **orthonormal frame bundle** of *E*, denoted F_{O}(*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 F_{SO}(*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 F_{O}*M* 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 F_{SO}*M*.

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 bundle map. One says that F_{O}(*E*) is a reduction of the structure group of F_{GL}(*E*) from GL(*k*, **R**) to O(*k*).

*G*-structures

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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 F_{GL}(*M*) to the orthogonal group O(*n*).

In general, if *M* is a smooth *n*-manifold and *G* is a Lie subgroup of GL(*n*, **R**) we define a ** G-structure** on

*M*to be a reduction of the structure group of F

_{GL}(

*M*) to

*G*. Explicitly, this is a principal

*G*-bundle F

_{G}(

*M*) over

*M*together with a

*G*-equivariant bundle map

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 2
*n*-dimensional symplectic manifold has a natural Sp(2*n*,**R**)-structure. - A 2
*n*-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 integrability condition is needed. A Sp(2*n*, **R**)-structure on *M* uniquely determines a nondegenerate 2-form on *M*, but for *M* to be symplectic, this 2-form must also be closed.

## References

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