Frame bundle

Summary

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 ).

The orthonormal frame bundle of the Möbius strip is a non-trivial principal -bundle over the circle.

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

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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 group   acts on   on the right as above. This action is clearly free and the orbits are just the fibers of  .

Principal bundle structure

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The frame bundle   can be given a natural topology and bundle structure determined by that of  . Let   be a local trivialization of  . Then for each xUi one has a linear isomorphism  . This data determines a bijection

 

given by

 

With these bijections, each   can be given the topology of  . The topology on   is the final topology coinduced by the inclusion maps  .

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  .

Associated vector bundles

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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   in  . Denote the equivalence classes by  .

The vector bundle   is naturally isomorphic to the bundle   where   is the fundamental representation of   on  . The isomorphism is given by

 

where   is a vector in   and   is a frame at  . One can easily check that this map is well-defined.

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.

Tangent frame bundle

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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  .

Smooth frames

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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.

Solder form

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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   at  . The solder form of   is the  -valued 1-form   defined by

 

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.

Orthonormal frame bundle

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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  .

G-structures

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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   to be a reduction of the structure group of   to  . Explicitly, this is a principal  -bundle   over   together with a  -equivariant bundle map

 

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  -structure on  .
  • A volume form on   determines a  -structure on  .
  • A  -dimensional symplectic manifold has a natural  -structure.
  • A  -dimensional complex or almost complex manifold has a natural  -structure.

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.

References

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  • 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