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2014-08-11T14:11:43Z

LucyMacCormick:

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 basis|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]]
:<math>p : \mathbf{R}^k \to E_x.</math>
The set of all frames at ''x'', denoted ''F''''x'', has a natural [[group action|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 [[Function composition|composition]] to give a new frame
:<math>p\circ g:\mathbf{R}^k\to E_x.</math>
This action of GL(''k'', '''R''') on ''F''''x'' 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, ''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 FGL(''E''), is the [[disjoint union]] of all the ''F''''x'':
:<math>\mathrm F(E) = \coprod_{x\in X}F_x.</math>
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 [[orbit (group theory)|orbit]]s 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
:<math>\psi_i : \pi^{-1}(U_i)\to U_i\times \mathrm{GL}(k, \mathbf R)</math>
given by
:<math>\psi_i(x,p) = (x,\varphi_{i,x}\circ p).</math>
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 map|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 bundle]]s. 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
:<math>\mathrm F(E)\times_{\rho}V</math>
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
:<math>[p,v]\mapsto p(v)</math>
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 representation|dual]] of the fundamental representation. [[Tensor bundle]]s 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===
[[Section (fiber bundle)|Local section]]s of the frame bundle of ''M'' are called [[smooth frame]]s 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
:<math>\psi(p) = (x, s(x)^{-1}\circ p)</math>
where ''p'' is a frame at ''x''. It follows that a manifold is [[Parallelizable manifold|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
:<math>\left(\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^n}\right)</math>
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 differential form|vector-valued 1-form]] on F''M'' called the '''[[solder form]]''' (also known as the '''fundamental''' or [[tautological one-form|'''tautological''' 1-form]]). Let ''x'' be a point of the manifold ''M'' and ''p'' a frame at ''x'', so that
:<math>p : \mathbf{R}^n\to T_xM</math>
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
:<math>\theta_p(\xi) = p^{-1}\mathrm d\pi(\xi)</math>
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 [[pushforward (differential)|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 [[equivariant|right equivariant]] in the sense that
:<math>R_g^*\theta = g^{-1}\theta</math>
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 map]]s ''TM'' → ''TM'' over ''M''. Viewed in this light θ is just the [[identity function|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 frame]]s for ''E''''x''. An orthonormal frame for ''E''''x'' is an ordered [[orthonormal basis]] for ''E''''x'', or, equivalently, a [[linear isometry]]
:<math>p:\mathbf{R}^k \to E_x</math>
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 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 [[orientability|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 FO''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 FSO''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
:<math>i:{\mathrm F}_{\mathrm O}(E) \to {\mathrm F}_{\mathrm{GL}}(E)</math>
is principal [[bundle map]]. One says that FO(''E'') is a [[reduction of the structure group]] of FGL(''E'') from GL(''k'', '''R''') to O(''k'').

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

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 [[Lie subgroup]] of GL(''n'', '''R''') we define a '''[[G-structure|''G''-structure]]''' on ''M'' to be a [[reduction of the structure group]] of FGL(''M'') to ''G''. Explicitly, this is a principal ''G''-bundle F''G''(''M'') over ''M'' together with a ''G''-equivariant [[bundle map]]
:<math>{\mathrm F}_{G}(M) \to {\mathrm F}_{\mathrm{GL}}(M)</math>
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 [[orientability|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 manifold|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 form|nondegenerate]] [[2-form]] on ''M'', but for ''M'' 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=Vol. 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}}
*{{Citation | last = Sternberg | first = S. | 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]]

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