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In mathematics, a **frame bundle** is a principal fiber bundle associated with any vector bundle *. The fiber of over a point ** is the set of all ordered bases, or **frames*, for *. The general linear group acts naturally on via a change of basis, giving the frame bundle the structure of a principal **-bundle (where **k* is the rank of *).
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The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the **tangent frame bundle**.

Let * * be a real vector bundle of rank * * over a topological space * *. A **frame** at a point * * is an ordered basis for the vector space * *. Equivalently, a frame can be viewed as a linear isomorphism

The set of all frames at * *, denoted * *, has a natural right action by the general linear group * * of invertible * * matrices: a group element * * acts on the frame * * via composition to give a new frame

This action of * * on * * 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, * * is homeomorphic to * * although it lacks a group structure, since there is no "preferred frame". The space * * is said to be a * *-torsor.

The **frame bundle** of * *, denoted by or , is the disjoint union of all the * *:

Each point in is a pair (*x*, *p*) where * * is a point in * * and * * is a frame at * *. There is a natural projection which sends * * to

The frame bundle can be given a natural topology and bundle structure determined by that of * *. Let * * be a local trivialization of

given by

With these bijections, each * * can be given the topology of

With all of the above data the frame bundle becomes a principal fiber bundle over * * with structure group * * and local trivializations * *. One can check that the transition functions of are the same as those of

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

A vector bundle * * and its frame bundle are associated bundles. Each one determines the other. The frame bundle can be constructed from * * as above, or more abstractly using the fiber bundle construction theorem. With the latter method, is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as * * but with abstract fiber * *, where the action of structure group * * on the fiber * * is that of left multiplication.

Given any linear representation * * there is a vector bundle

associated with which is given by product modulo the equivalence relation * * for all

The vector bundle * * is naturally isomorphic to the bundle where * * is the fundamental representation of

where * * is a vector in

Any vector bundle associated with * * can be given by the above construction. For example, the dual bundle of * * is given by where is the dual of the fundamental representation. Tensor bundles of * * can be constructed in a similar manner.

The **tangent frame bundle** (or simply the **frame bundle**) of a smooth manifold * * is the frame bundle associated with the tangent bundle of * *. The frame bundle of * * is often denoted * * or * * rather than * *. In physics, it is sometimes denoted * *. If * * is * *-dimensional then the tangent bundle has rank * *, so the frame bundle of * * is a principal * * bundle over * *.

Local sections of the frame bundle of * * are called smooth frames on * *. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in * * in * * which admits a smooth frame. Given a smooth frame * *, the trivialization * * is given by

where * * is a frame at * *. It follows that a manifold is parallelizable if and only if the frame bundle of * * admits a global section.

Since the tangent bundle of * * is trivializable over coordinate neighborhoods of * * so is the frame bundle. In fact, given any coordinate neighborhood * * with coordinates * * the coordinate vector fields

define a smooth frame on * *. 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.

The frame bundle of a manifold * * is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of * *. 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). Let * * be a point of the manifold * * and * * a frame at * *, so that

is a linear isomorphism of * * with the tangent space of

where ξ is a tangent vector to * * at the point * *, and * * is the inverse of the frame map, and * * is the differential of the projection map * *. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of * * and right equivariant in the sense that

where * * is right translation by * *. A form with these properties is called a basic or tensorial form on * *. Such forms are in 1-1 correspondence with * *-valued 1-forms on * * which are, in turn, in 1-1 correspondence with smooth bundle maps * * over * *. Viewed in this light * * is just the identity map on * *.

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.

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

where * * is equipped with the standard Euclidean metric. The orthogonal group * * 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 * *-torsor.

The **orthonormal frame bundle** of * *, denoted * *, is the set of all orthonormal frames at each point * * in the base space * *. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank * * Riemannian vector bundle * * is a principal * *-bundle over * *. Again, the construction works just as well in the smooth category.

If the vector bundle * * is orientable then one can define the **oriented orthonormal frame bundle** of * *, denoted * *, as the principal * *-bundle of all positively oriented orthonormal frames.

If * * is an * *-dimensional Riemannian manifold, then the orthonormal frame bundle of * *, denoted * * or * *, is the orthonormal frame bundle associated with the tangent bundle of * * (which is equipped with a Riemannian metric by definition). If * * is orientable, then one also has the oriented orthonormal frame bundle * *.

Given a Riemannian vector bundle * *, the orthonormal frame bundle is a principal * *-subbundle of the general linear frame bundle. In other words, the inclusion map

is principal bundle map. One says that * * is a reduction of the structure group of * * from * * to * *.

If a smooth manifold * * comes with additional structure it is often natural to consider a subbundle of the full frame bundle of * * which is adapted to the given structure. For example, if * * 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 * * to the orthogonal group * *.

In general, if * * is a smooth * *-manifold and * * is a Lie subgroup of * * we define a ** G-structure** on

over * *.

In this language, a Riemannian metric on * * gives rise to an * *-structure on * *. The following are some other examples.

- Every oriented manifold has an oriented frame bundle which is just a
- A volume form on
- A
- A

In many of these instances, a * *-structure on * * uniquely determines the corresponding structure on * *. For example, a * *-structure on * * determines a volume form on * *. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A * *-structure on * * uniquely determines a nondegenerate 2-form on * *, but for * * to be symplectic, this 2-form must also be closed.

- Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
*Foundations of Differential Geometry*, vol. 1 (New ed.), Wiley Interscience, 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 - Sternberg, S. (1983),
*Lectures on Differential Geometry*((2nd ed.) ed.), New York: Chelsea Publishing Co., ISBN 0-8218-1385-4