In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.

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 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

 
 
 
 
 

Connection (mathematics)
Differential geometry
Maps of manifolds
Smooth functions