In differential geometry, the classical notion of a connection, usually encountered as a structure formalising the idea of parallel transport on a vector bundle, or more generally on a fiber bundle, can be further generalised to that of a connection on a fibered manifold. A fibered manifold is a surjective submersion of smooth manifolds . Since fiber bundles are simply locally trivial fibered manifolds, the notion of a connection as discussed in this article provides a general framework which includes the more classical meaning.
Formal definition Let be a fibered manifold. A generalized connection on is a section , where is the jet manifold of .
Connection as a horizontal splitting With the above manifold there is the following canonical short exact sequence of vector bundles over :
where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto .
A connection on a fibered manifold is defined as a linear bundle morphism
over which splits the exact sequence . A connection always exists.
Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution
of and its horizontal decomposition .
At the same time, by an Ehresmann connection also is meant the following construction. Any connection on a fibered manifold yields a horizontal lift of a vector field on onto , but need not defines the similar lift of a path in into . Let
be two smooth paths in and , respectively. Then is called the horizontal lift of if
A connection is said to be the Ehresmann connection if, for each path in , there exists its horizontal lift through any point . A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Connection as a tangent-valued form Given a fibered manifold , let it be endowed with an atlas of fibered coordinates , and let be a connection on . It yields uniquely the horizontal tangent-valued one-form
on which projects onto the canonical tangent-valued form (tautological one-form or solder form)
on , and vice versa. With this form, the horizontal splitting reads
In particular, the connection in yields the horizontal lift of any vector field on to a projectable vector field
on .
Connection as a vertical-valued form The horizontal splitting of the exact sequence defines the corresponding splitting of the dual exact sequence
where and are the cotangent bundles of , respectively, and is the dual bundle to , called the vertical cotangent bundle. This splitting is given by the vertical-valued form
which also represents a connection on a fibered manifold.
Treating a connection as a vertical-valued form, one comes to the following important construction. Given a fibered manifold , let be a morphism and the pullback bundle of by . Then any connection on induces the pullback connection
on .
Connection as a jet bundle section Let be the jet manifold of sections of a fibered manifold , with coordinates . Due to the canonical imbedding
any connection on a fibered manifold is represented by a global section
of the jet bundle , and vice versa. It is an affine bundle modelled on a vector bundle
There are the following corollaries of this fact.
Curvature and torsion Given the connection on a fibered manifold , its curvature is defined as the Nijenhuis differential
This is a vertical-valued horizontal two-form on .
Given the connection and the soldering form , a torsion of with respect to is defined as
Bundle of principal connections Let be a principal bundle with a structure Lie group . A principal connection on usually is described by a Lie algebra-valued connection one-form on . At the same time, a principal connection on is a global section of the jet bundle which is equivariant with respect to the canonical right action of in . Therefore, it is represented by a global section of the quotient bundle , called the bundle of principal connections. It is an affine bundle modelled on the vector bundle whose typical fiber is the Lie algebra of structure group , and where acts on fibers by the adjoint representation. There is the canonical imbedding of to the quotient bundle which also is called the bundle of principal connections.
Given a basis } for a Lie algebra of , the fiber bundle is endowed with bundle coordinates , and its sections are represented by vector-valued one-forms
where
are the familiar local connection forms on .
Let us note that the jet bundle of is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
where
is called the strength form of a principal connection.
See also Connection (mathematics) Fibred manifold Ehresmann connection Connection (principal bundle)
Notes
References