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*).

- Definition and construction
- Associated vector bundles
- Tangent frame bundle
- Smooth frames
- Solder form
- Orthonormal frame bundle
- G-structures
- References

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**.

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*.

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.

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*.

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.

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*), and *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*.

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 *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*).

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

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.

In mathematics, a **vector bundle** is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space *X* : to every point *x* of the space *X* we associate a vector space *V*(*x*) in such a way that these vector spaces fit together to form another space of the same kind as *X*, which is then called a **vector bundle over X**.

In mathematics, a **principal bundle** is a mathematical object that formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) ↦*x*.

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the **spinor bundle** to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors .

In mathematics, and especially differential geometry and gauge theory, a **connection** on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In mathematics, a **moving frame** is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

In the branch of mathematics called differential geometry, an **affine connection** is a geometric object on a smooth manifold which *connects* nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space **R**^{n} by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

In differential geometry, a ** G-structure** on an

In mathematics, the **Stiefel manifold** is the set of all orthonormal *k*-frames in That is, it is the set of ordered orthonormal *k*-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold of orthonormal *k*-frames in and the quaternionic Stiefel manifold of orthonormal *k*-frames in . More generally, the construction applies to any real, complex, or quaternionic inner product space.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In differential geometry, a **spin structure** on an orientable Riemannian manifold (*M*, *g*) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.

In differential geometry, a branch of mathematics, a **Riemannian submersion** is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.

In differential geometry, given a spin structure on an *n*-dimensional orientable Riemannian manifold, a section of the spinor bundle **S** is called a **spinor field**. A spinor bundle is the complex vector bundle associated to the corresponding principal bundle of spin frames over *M* via the spin representation of its structure group Spin(*n*) on the space of spinors Δ_{n}.

In differential geometry, an **Ehresmann connection** is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.

In Riemannian geometry, the **unit tangent bundle** of a Riemannian manifold (*M*, *g*), denoted by T^{1}*M*, UT(*M*) or simply UT*M*, is the unit sphere bundle for the tangent bundle T(*M*). It is a fiber bundle over *M* whose fiber at each point is the unit sphere in the tangent bundle:

In mathematics, a **Clifford bundle** is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold *M* which is called the Clifford bundle of *M*.

In mathematics, more precisely in differential geometry, a **soldering** of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950.

In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a **bundle metric**, or **fibre metric**.

In differential geometry, the **Kosmann lift**, named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natural lift defined on the bundle of linear frames.

In symplectic geometry, the **symplectic frame bundle** of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic with respect to . In other words, an element of the **symplectic frame bundle** is a linear frame at point i.e. an ordered basis of tangent vectors at of the tangent vector space , satisfying

- 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

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