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 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 ', 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 '.
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 ' 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 '.
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 defining 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.
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 ' at '. The solder form of ' is the '-valued 1-form ' defined by
where þ 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 ' 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.
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.