Editing Frame bundle

{{Short description|Functions in mathematics}}
[[File:Mobius frame bundle.png|thumb|The orthonormal frame bundle <math>\mathcal{F_O}(E)</math> of the [[Möbius strip]] <math>E</math> is a non-trivial principal <math>\mathbb{Z}/2\mathbb{Z}</math>-bundle over the circle.]]
In [[mathematics]], a '''frame bundle''' is a [[principal fiber bundle]] <math>F(E)</math> associated with any [[vector bundle]] ''<math>E</math>''. The fiber of <math>F(E)</math> over a point ''<math>x</math>'' is the set of all [[ordered basis|ordered bases]], or ''frames'', for ''<math>E_x</math>''. The [[general linear group]] acts naturally on <math>F(E)</math> via a [[change of basis]], giving the frame bundle the structure of a principal ''<math>\mathrm{GL}(k,\mathbb{R})</math>''-bundle (where ''k'' is the rank of ''<math>E</math>'').

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'''.

==Definition and construction==
Let ''<math>E \to X</math>'' be a real [[vector bundle]] of rank ''<math>k</math>'' over a [[topological space]] ''<math>X</math>''. A '''frame''' at a point ''<math>x \in X</math>'' is an [[ordered basis]] for the vector space ''<math>E_x</math>''. Equivalently, a frame can be viewed as a [[linear isomorphism]]
:<math>p : \mathbf{R}^k \to E_x.</math>
The set of all frames at ''<math>x</math>'', denoted ''<math>F_x</math>'', has a natural [[Group action (mathematics)|right action]] by the [[general linear group]] ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' of invertible ''<math>k \times k</math>'' matrices: a group element ''<math>g \in \mathrm{GL}(k,\mathbb{R})</math>'' acts on the frame ''<math>p</math>'' via [[Function composition|composition]] to give a new frame
:<math>p\circ g:\mathbf{R}^k\to E_x.</math>
This action of ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on ''<math>F_x</math>'' is both [[free action|free]] and [[transitive action|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, ''<math>F_x</math>'' is [[homeomorphic]] to ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' although it lacks a group structure, since there is no "preferred frame". The space ''<math>F_x</math>'' is said to be a ''<math>\mathrm{GL}(k,\mathbb{R})</math>''-[[torsor]].

The '''frame bundle''' of ''<math>E</math>'', denoted by <math>F(E)</math> or <math>F_{\mathrm{GL}}(E)</math>, is the [[disjoint union]] of all the ''<math>F_x</math>'':
:<math>\mathrm F(E) = \coprod_{x\in X}F_x.</math>
Each point in <math>F(E)</math> is a pair (''x'', ''p'') where ''<math>x</math>'' is a point in ''<math>X</math>'' and ''<math>p</math>'' is a frame at ''<math>x</math>''. There is a natural projection <math>\pi: F(E)\to X</math> which sends '''''<math>(x,p)</math>''''' to ''<math>x</math>''. The group ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' acts on <math>F(E)</math> on the right as above. This action is clearly free and the [[orbit (group theory)|orbit]]s are just the fibers of '''''<math>\pi</math>'''''.

=== Principal bundle structure ===
The frame bundle <math>F(E)</math> can be given a natural topology and bundle structure determined by that of ''<math>E</math>''. Let '''''<math>(U_i,\phi_i)</math>''''' be a [[local trivialization]] of ''<math>E</math>''. Then for each ''x'' ∈ ''U''<sub>''i''</sub> one has a linear isomorphism '''''<math>\phi_{i,x}: E_x \to \mathbb{R}^k</math>'''''. This data determines a bijection
:<math>\psi_i : \pi^{-1}(U_i)\to U_i\times \mathrm{GL}(k, \mathbb{R})</math>
given by
:<math>\psi_i(x,p) = (x,\phi_{i,x}\circ p).</math>
With these bijections, each '''''<math>\pi^{-1}(U_i)</math>''''' can be given the topology of ''<math>U_i \times \mathrm{GL}(k,\mathbb{R})</math>''. The topology on <math>F(E)</math> is the [[final topology]] coinduced by the inclusion maps '''''<math>\pi^{-1}(U_i) \to F(E)</math>'''''.

With all of the above data the frame bundle <math>F(E)</math> becomes a [[principal fiber bundle]] over ''<math>X</math>'' with [[structure group]] ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' and local trivializations '''''<math>(\{U_i\},\{\psi_i\})</math>'''''. One can check that the [[Transition map|transition functions]] of <math>F(E)</math> are the same as those of ''<math>E</math>''.

The above all works in the smooth category as well: if ''<math>E</math>'' is a smooth vector bundle over a [[smooth manifold]] ''<math>M</math>'' then the frame bundle of ''<math>E</math>'' can be given the structure of a smooth principal bundle over ''<math>M</math>''.

==Associated vector bundles==
A vector bundle ''<math>E</math>'' and its frame bundle <math>F(E)</math> are [[associated bundle]]s. Each one determines the other. The frame bundle <math>F(E)</math> can be constructed from ''<math>E</math>'' as above, or more abstractly using the [[fiber bundle construction theorem]]. With the latter method, <math>F(E)</math> is the fiber bundle with same base, structure group, trivializing neighborhoods, and transition functions as ''<math>E</math>'' but with abstract fiber ''<math>\mathrm{GL}(k,\mathbb{R})</math>'', where the action of structure group ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on the fiber ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' is that of left multiplication.

Given any [[linear representation]] ''<math>\rho: \mathrm{GL}(k,\mathbb{R}) \to \mathrm{GL}(V,\mathbb{F})</math>'' there is a vector bundle
:<math>\mathrm F(E)\times_{\rho}V</math>
associated with <math>F(E)</math> which is given by product <math>F(E) \times V</math> modulo the [[equivalence relation]] '''''<math>(pg,v) \sim (p, \rho(g)v)</math>''''' for all ''<math>g</math>'' in ''<math>\mathrm{GL}(k,\mathbb{R})</math>''. Denote the equivalence classes by '''''<math>[p,v]</math>'''''.

The vector bundle ''<math>E</math>'' is [[naturally isomorphic]] to the bundle <math>F(E) \times_\rho \mathbb{R}^k</math> where '''''<math>\rho</math>''''' is the fundamental representation of ''<math>\mathrm{GL}(k,\mathbb{R})</math>'' on '''''<math>\mathbb{R}^k</math>'''''. The isomorphism is given by
:<math>[p,v]\mapsto p(v)</math>
where '''''<math>v</math>''''' is a vector in '''''<math>\mathbb{R}^k</math>''''' and '''''<math>p: \mathbb{R}^k \to E_x</math>''''' is a frame at ''<math>x</math>''. One can easily check that this map is [[well-defined]].

Any vector bundle associated with ''<math>E</math>'' can be given by the above construction. For example, the [[dual bundle]] of ''<math>E</math>'' is given by <math>F(E) \times_{\rho^*} (\mathbb{R}^k)^*</math> where <math>\rho^*</math> is the [[dual representation|dual]] of the fundamental representation. [[Tensor bundle]]s of ''<math>E</math>'' can be constructed in a similar manner.

==Tangent frame bundle==
The '''tangent frame bundle''' (or simply the '''frame bundle''') of a [[smooth manifold]] ''<math>M</math>'' is the frame bundle associated with the [[tangent bundle]] of ''<math>M</math>''. The frame bundle of ''<math>M</math>'' is often denoted ''<math>FM</math>'' or ''<math>\mathrm{GL}(M)</math>'' rather than ''<math>F(TM)</math>''. In physics, it is sometimes denoted ''<math>LM</math>''. If ''<math>M</math>'' is ''<math>n</math>''-dimensional then the tangent bundle has rank ''<math>n</math>'', so the frame bundle of ''<math>M</math>'' is a principal ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' bundle over ''<math>M</math>''.

===Smooth frames===
[[Section (fiber bundle)|Local section]]s of the frame bundle of ''<math>M</math>'' are called [[smooth frame]]s on ''<math>M</math>''. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in ''<math>U</math>'' in ''<math>M</math>'' which admits a smooth frame. Given a smooth frame ''<math>s: U \to FU</math>'', the trivialization ''<math>\psi: FU \to U \times \mathrm{GL}(n,\mathbb{R})</math>'' is given by
:<math>\psi(p) = (x, s(x)^{-1}\circ p)</math>
where ''<math>p</math>'' is a frame at ''<math>x</math>''. It follows that a manifold is [[Parallelizable manifold|parallelizable]] if and only if the frame bundle of ''<math>M</math>'' admits a global section.

Since the tangent bundle of ''<math>M</math>'' is trivializable over coordinate neighborhoods of ''<math>M</math>'' so is the frame bundle. In fact, given any coordinate neighborhood ''<math>U</math>'' with coordinates ''<math>(x^1,\ldots,x^n)</math>'' the coordinate vector fields
:<math>\left(\frac{\partial}{\partial x^1},\ldots,\frac{\partial}{\partial x^n}\right)</math>
define a smooth frame on ''<math>U</math>''. 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 ''<math>M</math>'' is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of ''<math>M</math>''. This relationship can be expressed by means of a [[vector-valued differential form|vector-valued 1-form]] on ''<math>FM</math>'' called the '''[[solder form]]''' (also known as the '''fundamental''' or [[tautological one-form|'''tautological''' 1-form]]). Let ''<math>x</math>'' be a point of the manifold ''<math>M</math>'' and ''<math>p</math>'' a frame at ''<math>x</math>'', so that
:<math>p : \mathbf{R}^n\to T_xM</math>
is a linear isomorphism of '''''<math>\mathbb{R}^n</math>''''' with the tangent space of ''<math>M</math>'' at ''<math>x</math>''. The solder form of ''<math>FM</math>'' is the '''''<math>\mathbb{R}^n</math>'''''-valued 1-form ''<math>\theta</math>'' defined by
:<math>\theta_p(\xi) = p^{-1}\mathrm d\pi(\xi)</math>
where ξ is a tangent vector to ''<math>FM</math>'' at the point ''<math>(x,p)</math>'', and ''<math>p^{-1}: T_x M \to \mathbb{R}^n</math>'' is the inverse of the frame map, and ''<math>d\pi</math>'' is the [[pushforward (differential)|differential]] of the projection map ''<math>\pi: FM \to M</math>''. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of ''<math>\pi</math>'' and [[equivariant|right equivariant]] in the sense that
:<math>R_g^*\theta = g^{-1}\theta</math>
where ''<math>R_g</math>'' is right translation by ''<math>g \in \mathrm{GL}(n,\mathbb{R})</math>''. A form with these properties is called a basic or [[tensorial form]] on ''<math>FM</math>''. Such forms are in 1-1 correspondence with ''<math>TM</math>''-valued 1-forms on ''<math>M</math>'' which are, in turn, in 1-1 correspondence with smooth [[bundle map]]s ''<math>TM \to TM</math>'' over ''<math>M</math>''. Viewed in this light ''<math>\theta</math>'' is just the [[identity function|identity map]] on ''<math>TM</math>''.

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 ''<math>E</math>'' is equipped with a [[Riemannian bundle metric]] then each fiber ''<math>E_x</math>'' is not only a vector space but an [[inner product space]]. It is then possible to talk about the set of all [[orthonormal frame]]s for ''<math>E_x</math>''. An orthonormal frame for ''<math>E_x</math>'' is an ordered [[orthonormal basis]] for ''<math>E_x</math>'', or, equivalently, a [[linear isometry]]
:<math>p:\mathbb{R}^k \to E_x</math>
where ''<math>\mathbb{R}^k</math>'' is equipped with the standard [[Euclidean metric]]. The [[orthogonal group]] ''<math>\mathrm{O}(k)</math>'' 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 ''<math>\mathrm{O}(k)</math>''-[[torsor]].

The '''orthonormal frame bundle''' of ''<math>E</math>'', denoted ''<math>F_{\mathrm{O}}(E)</math>'', is the set of all orthonormal frames at each point ''<math>x</math>'' in the base space ''<math>X</math>''. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. The orthonormal frame bundle of a rank ''<math>k</math>'' Riemannian vector bundle ''<math>E \to X</math>'' is a principal ''<math>\mathrm{O}(k)</math>''-bundle over ''<math>X</math>''. Again, the construction works just as well in the smooth category.

If the vector bundle ''<math>E</math>'' is [[orientability|orientable]] then one can define the '''oriented orthonormal frame bundle''' of ''<math>E</math>'', denoted ''<math>F_{\mathrm{SO}}(E)</math>'', as the principal ''<math>\mathrm{SO}(k)</math>''-bundle of all positively oriented orthonormal frames.

If ''<math>M</math>'' is an ''<math>n</math>''-dimensional [[Riemannian manifold]], then the orthonormal frame bundle of ''<math>M</math>'', denoted ''<math>F_{\mathrm{O}}(M)</math>'' or ''<math>\mathrm{O} (M)</math>'', is the orthonormal frame bundle associated with the tangent bundle of ''<math>M</math>'' (which is equipped with a Riemannian metric by definition). If ''<math>M</math>'' is orientable, then one also has the oriented orthonormal frame bundle ''<math>F_{\mathrm{SO}}M</math>''.

Given a Riemannian vector bundle ''<math>E</math>'', the orthonormal frame bundle is a principal ''<math>\mathrm{O}(k)</math>''-[[subbundle]] of the general linear frame bundle. In other words, the inclusion map
:<math>i:{\mathrm F}_{\mathrm O}(E) \to {\mathrm F}_{\mathrm{GL}}(E)</math>
is principal [[bundle map]]. One says that ''<math>F_{\mathrm{O}}(E)</math>'' is a [[reduction of the structure group]] of ''<math>F_{\mathrm{GL}}(E)</math>'' from ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' to ''<math>\mathrm{O}(k)</math>''.

==''G''-structures==
{{see also|G-structure}}

If a smooth manifold ''<math>M</math>'' comes with additional structure it is often natural to consider a subbundle of the full frame bundle of ''<math>M</math>'' which is adapted to the given structure. For example, if ''<math>M</math>'' is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of ''<math>M</math>''. The orthonormal frame bundle is just a reduction of the structure group of ''<math>F_{\mathrm{GL}}(M)</math>'' to the orthogonal group ''<math>\mathrm{O}(n)</math>''.

In general, if ''<math>M</math>'' is a smooth ''<math>n</math>''-manifold and ''<math>G</math>'' is a [[Lie subgroup]] of ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' we define a '''[[G-structure|''G''-structure]]''' on ''<math>M</math>'' to be a [[reduction of the structure group]] of ''<math>F_{\mathrm{GL}}(M)</math>'' to ''<math>G</math>''. Explicitly, this is a principal ''<math>G</math>''-bundle ''<math>F_{G}(M)</math>'' over ''<math>M</math>'' together with a ''<math>G</math>''-equivariant [[bundle map]]
:<math>{\mathrm F}_{G}(M) \to {\mathrm F}_{\mathrm{GL}}(M)</math>
over ''<math>M</math>''.

In this language, a Riemannian metric on ''<math>M</math>'' gives rise to an ''<math>\mathrm{O}(n)</math>''-structure on ''<math>M</math>''. The following are some other examples.

*Every [[orientability|oriented manifold]] has an oriented frame bundle which is just a ''<math>\mathrm{GL}^+(n,\mathbb{R})</math>''-structure on ''<math>M</math>''.
*A [[volume form]] on ''<math>M</math>'' determines a ''<math>\mathrm{SL}(n,\mathbb{R})</math>''-structure on ''<math>M</math>''.
*A ''<math>2n</math>''-dimensional [[symplectic manifold]] has a natural ''<math>\mathrm{Sp}(2n,\mathbb{R})</math>''-structure.
*A ''<math>2n</math>''-dimensional [[complex manifold|complex]] or [[almost complex manifold]] has a natural ''<math>\mathrm{GL}(n,\mathbb{C})</math>''-structure.
In many of these instances, a ''<math>G</math>''-structure on ''<math>M</math>'' uniquely determines the corresponding structure on ''<math>M</math>''. For example, a ''<math>\mathrm{SL}(n,\mathbb{R})</math>''-structure on ''<math>M</math>'' determines a volume form on ''<math>M</math>''. However, in some cases, such as for symplectic and complex manifolds, an added [[integrability condition]] is needed. A ''<math>\mathrm{Sp}(2n,\mathbb{R})</math>''-structure on ''<math>M</math>'' uniquely determines a [[nondegenerate form|nondegenerate]] [[2-form]] on ''<math>M</math>'', but for ''<math>M</math>'' to be symplectic, this 2-form must also be [[closed differential form|closed]].

==References==
* {{citation | last1=Kobayashi|first1=Shoshichi|last2=Nomizu|first2=Katsumi | title = [[Foundations of Differential Geometry]]|volume=1| publisher=[[Wiley Interscience]] | year=1996|edition=New|isbn=0-471-15733-3}}
* {{citation|last1=Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year=1993|publisher=Springer-Verlag|access-date=2008-08-02|archive-url=https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf|archive-date=2017-03-30|url-status=dead}}
*{{Citation | last = Sternberg | first = S. |authorlink = Shlomo Sternberg | year = 1983 | title = Lectures on Differential Geometry | edition = (2nd ed.) | publisher = Chelsea Publishing Co. | location = New York | isbn = 0-8218-1385-4}}

[[Category:Fiber bundles]]
[[Category:Vector bundles]]