Connection (fibred Manifold)

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

: $\mathrmY=\Gamma\left(Y\times_X \mathrmX \right) \subset \mathrmY$

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

:\begin\mathbb R\supset ni t&\to x(t)\in X \\ \mathbb R\ni t&\to y(t)\in Y\end

be two smooth paths in and , respectively. Then is called the horizontal lift of if

:$\pi(y(t))= x(t)\,, \qquad \dot y(t)\in \mathrmY \,, \qquad t\in\mathbb R\,.$

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)

: $\theta_X=dx^\mu\otimes\partial_\mu$

on , and ''vice versa''. With this form, the horizontal splitting reads

: $\Gamma:\partial_\mu\to \partial_\mu\rfloor\Gamma=\partial_\mu +\Gamma^i_\mu\partial_i\,.$

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

:$\Gamma \tau=\tau\rfloor\Gamma=\tau^\mu\left(\partial_\mu +\Gamma^i_\mu\partial_i\right)\subset \mathrmY$

on .

Connection as a vertical-valued form

The horizontal splitting of the exact sequence defines the corresponding splitting of the dual exact sequence

: $0\to Y\times_X \mathrm^*X \to \mathrm^*Y\to \mathrm^*Y\to 0\,,$

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

: $\Gamma= \left(dy^i -\Gamma^i_\lambda dx^\lambda\right)\otimes\partial_i\,,$

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''

: $f*\Gamma=\left(dy^i-\left(\Gamma\circ \tilde f\right)^i_\lambda\fracdx'^\mu\right)\otimes\partial_i$

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

: $\mathrm^1Y\to_Y \left(Y\times_X \mathrm^*X \right)\otimes_Y \mathrmY\,, \qquad \left(y^i_\mu\right)\to dx^\mu\otimes \left(\partial_\mu + y^i_\mu\partial_i\right)\,,$

any connection on a fibered manifold is represented by a global section

: $\Gamma :Y\to \mathrm^1Y\,, \qquad y_\lambda^i\circ\Gamma=\Gamma_\lambda^i\,,$

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

: \begin
R&=\tfrac12 d_\Gamma\Gamma\\&=\tfrac12 Gamma,\Gamma\mathrm \\&= \tfrac12 R_^i \, dx^\lambda\wedge dx^\mu\otimes\partial_i\,, \\
R_^i &= \partial_\lambda\Gamma_\mu^i - \partial_\mu\Gamma_\lambda^i + \Gamma_\lambda^j\partial_j \Gamma_\mu^i - \Gamma_\mu^j\partial_j \Gamma_\lambda^i\,.
\end

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

: $T = d_\Gamma \sigma = \left(\partial_\lambda\sigma_\mu^i + \Gamma_\lambda^j\partial_j\sigma_\mu^i -\partial_j\Gamma_\lambda^i\sigma_\mu^j\right) \, dx^\lambda\wedge dx^\mu\otimes \partial_i\,.$

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

: $A=dx^\lambda\otimes \left(\partial_\lambda + a^m_\lambda _m\right)\,,$

where
: $a^m_\lambda \, dx^\lambda\otimes _m$

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

: $\begin a_^r &= \tfrac12\left(F_^r + S_^r\right) \\ &= \tfrac12\left(a_^r + a_^r - c_^r a_\lambda^p a_\mu^q\right) + \tfrac12\left(a_^r - a_^r + c_^r a_\lambda^p a_\mu^q\right)\,, \end$

where

: $F=\tfrac F_^m \, dx^\lambda\wedge dx^\mu\otimes _m$

is called the ''strength form'' of a principal connection.

See also

*Connection (mathematics)
* Fibred manifold
* Ehresmann connection
*Connection (principal bundle)

Notes

References

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{{Manifolds

Connection (mathematics)
Differential geometry
Maps of manifolds
Smooth functions