Connection (fibred manifold)

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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 .[1]

Connection as a horizontal splitting

Let be a fibered 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 (1). 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 and be 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 trough 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) on , and vice versa. With this form, the horizontal splitting (2) reads

In particular, the connection (3) yields the horizontal lift of any vector field on to a projectable vector field

on .

Connection as a vertical-valued form

The horizontal splitting (2) of the exact sequence (1) 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 (3) 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 (3) 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.

(i) Connections on a fibered manifold make up an affine space modelled on the vector space of soldering forms

on , i.e., sections of the vector bundle (4).

(ii) Connection coefficients possess the coordinate transformation law

(iii) Every connection on a fibred manifold yields the first order differential operator

on called the covariant differential relative to the connection . If is a section, its covariant differential

and the covariant derivative along a vector field on are defined.

Curvature and torsion

Given the connection (3) on a fibered manifold , its curvature is defined as the Nijenhuis differential

This is a vertical-valued horizontal two-form on .

Given the connection (3) and the soldering form (5), 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


is called the strength form of a principal connection.

See also


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  • Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.

External links

  • Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886