Connection (fibred manifold)

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Template:Math. 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 Template:Math be a fibered manifold. A generalized connection on Template:Mvar is a section Template:Math, where Template:Math is the jet manifold of Template:Mvar.

Connection as a horizontal splitting

With the above manifold Template:Mvar there is the following canonical short exact sequence of vector bundles over Template:Mvar:

where Template:Math and Template:Math are the tangent bundles of Template:Mvar, respectively, Template:Math is the vertical tangent bundle of Template:Mvar, and Template:Math is the pullback bundle of Template:Math onto Template:Mvar.

A connection on a fibered manifold Template:Math is defined as a linear bundle morphism

over Template:Mvar which splits the exact sequence Template:EquationRef. A connection always exists.

Sometimes, this connection Template:Math is called the Ehresmann connection because it yields the horizontal distribution

$\mathrm{H}Y=\mathrm{\Gamma }\left(Y{\times }_X\mathrm{T}X\right)\subset \mathrm{T}Y$

of Template:Math and its horizontal decomposition Template:Math.

At the same time, by an Ehresmann connection also is meant the following construction. Any connection Template:Math on a fibered manifold Template:Math yields a horizontal lift Template:Math of a vector field Template:Mvar on Template:Mvar onto Template:Mvar, but need not defines the similar lift of a path in Template:Mvar into Template:Mvar. Let

$\begin{array}{cc}ℝ\supset [,]\ni t& \to x(t)\in X\\ ℝ\ni t& \to y(t)\in Y\end{array}$

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

$\pi (y(t))=x(t),\dot{y}(t)\in \mathrm{H}Y,t\in ℝ.$

A connection Template:Math is said to be the Ehresmann connection if, for each path Template:Math in Template:Mvar, there exists its horizontal lift through any point Template:Math. 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 Template:Math, let it be endowed with an atlas of fibered coordinates Template:Math, and let Template:Math be a connection on Template:Math. It yields uniquely the horizontal tangent-valued one-form

on Template:Mvar which projects onto the canonical tangent-valued form (tautological one-form or solder form)

${\theta }_X=d{x}^{\mu }\otimes {\partial }_{\mu }$

on Template:Mvar, and vice versa. With this form, the horizontal splitting Template:EquationNote reads

$\mathrm{\Gamma }:{\partial }_{\mu }\to {\partial }_{\mu }\rfloor \mathrm{\Gamma }={\partial }_{\mu }+{\mathrm{\Gamma }}_{\mu }^i{\partial }_i.$

In particular, the connection Template:Math in Template:EquationNote yields the horizontal lift of any vector field Template:Math on Template:Mvar to a projectable vector field

$\mathrm{\Gamma }\tau =\tau \rfloor \mathrm{\Gamma }={\tau }^{\mu }({\partial }_{\mu }+{\mathrm{\Gamma }}_{\mu }^i{\partial }_i)\subset \mathrm{H}Y$

on Template:Mvar.

Connection as a vertical-valued form

The horizontal splitting Template:EquationNote of the exact sequence Template:EquationNote defines the corresponding splitting of the dual exact sequence

$0\to Y{\times }_X{\mathrm{T}}^{*}X\to {\mathrm{T}}^{*}Y\to {\mathrm{V}}^{*}Y\to 0,$

where Template:Math and Template:Math are the cotangent bundles of Template:Mvar, respectively, and Template:Math is the dual bundle to Template:Math, called the vertical cotangent bundle. This splitting is given by the vertical-valued form

$\mathrm{\Gamma }=(d{y}^i-{\mathrm{\Gamma }}_{\lambda }^{i}d{x}^{\lambda })\otimes {\partial }_i,$

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 Template:Math, let Template:Math be a morphism and Template:Math the pullback bundle of Template:Mvar by Template:Mvar. Then any connection Template:Math Template:EquationNote on Template:Math induces the pullback connection

$f*\mathrm{\Gamma }=(d{y}^i-{(\mathrm{\Gamma }\circ \tilde{f})}_{\lambda }^i\frac{\partial f^{\lambda }}{\partial x{\text{'}}^{\mu }}dx{\text{'}}^{\mu })\otimes {\partial }_i$

on Template:Math.

Connection as a jet bundle section

Let Template:Math be the jet manifold of sections of a fibered manifold Template:Math, with coordinates Template:Math. Due to the canonical imbedding

${\mathrm{J}}^{1}Y{\to }_Y\left(Y{\times }_X{\mathrm{T}}^{*}X\right){\otimes }_Y\mathrm{T}Y,\left(y_{\mu }^i\right)\to d{x}^{\mu }\otimes ({\partial }_{\mu }+y_{\mu }^i{\partial }_i),$

any connection Template:Math Template:EquationNote on a fibered manifold Template:Math is represented by a global section

$\mathrm{\Gamma }:Y\to {\mathrm{J}}^{1}Y,y_{\lambda }^i\circ \mathrm{\Gamma }={\mathrm{\Gamma }}_{\lambda }^i,$

of the jet bundle Template:Math, 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 Template:Math Template:EquationNote on a fibered manifold Template:Math, its curvature is defined as the Nijenhuis differential

$\begin{array}{cc}R& =\frac{1}{2}{d}_{\mathrm{\Gamma }}\mathrm{\Gamma }\\ & =\frac{1}{2}[\mathrm{\Gamma },\mathrm{\Gamma }{]}_{\mathrm{F}\mathrm{N}}\\ & =\frac{1}{2}{R}_{\lambda \mu }^{i}d{x}^{\lambda }\wedge d{x}^{\mu }\otimes {\partial }_i,\\ R_{\lambda \mu }^i& ={\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu }^i-{\partial }_{\mu }{\mathrm{\Gamma }}_{\lambda }^i+{\mathrm{\Gamma }}_{\lambda }^j{\partial }_j{\mathrm{\Gamma }}_{\mu }^i-{\mathrm{\Gamma }}_{\mu }^j{\partial }_j{\mathrm{\Gamma }}_{\lambda }^i.\end{array}$

This is a vertical-valued horizontal two-form on Template:Mvar.

Given the connection Template:Math Template:EquationNote and the soldering form Template:Mvar Template:EquationNote, a torsion of Template:Math with respect to Template:Mvar is defined as

$T=d_{\mathrm{\Gamma }}\sigma =({\partial }_{\lambda }{\sigma }_{\mu }^i+{\mathrm{\Gamma }}_{\lambda }^j{\partial }_j{\sigma }_{\mu }^i-{\partial }_j{\mathrm{\Gamma }}_{\lambda }^i{\sigma }_{\mu }^j)d{x}^{\lambda }\wedge d{x}^{\mu }\otimes {\partial }_i.$

Bundle of principal connections

Let Template:Math be a principal bundle with a structure Lie group Template:Mvar. A principal connection on Template:Mvar usually is described by a Lie algebra-valued connection one-form on Template:Mvar. At the same time, a principal connection on Template:Mvar is a global section of the jet bundle Template:Math which is equivariant with respect to the canonical right action of Template:Mvar in Template:Mvar. Therefore, it is represented by a global section of the quotient bundle Template:Math, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle Template:Math whose typical fiber is the Lie algebra Template:Math of structure group Template:Mvar, and where Template:Mvar acts on by the adjoint representation. There is the canonical imbedding of Template:Mvar to the quotient bundle Template:Math which also is called the bundle of principal connections.

Given a basis Template:Math} for a Lie algebra of Template:Mvar, the fiber bundle Template:Mvar is endowed with bundle coordinates Template:Math, and its sections are represented by vector-valued one-forms

$A=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+a_{\lambda }^m{\mathrm{e}}_m),$

where

$a_{\lambda }^{m}d{x}^{\lambda }\otimes {\mathrm{e}}_m$

are the familiar local connection forms on Template:Mvar.

Let us note that the jet bundle Template:Math of Template:Mvar is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition

$\begin{array}{cc}{a}_{\lambda \mu }^r& =\frac{1}{2}(F_{\lambda \mu }^r+S_{\lambda \mu }^r)\\ & =\frac{1}{2}(a_{\lambda \mu }^r+a_{\mu \lambda }^r-c_{pq}^r{a}_{\lambda }^p{a}_{\mu }^q)+\frac{1}{2}(a_{\lambda \mu }^r-a_{\mu \lambda }^r+c_{pq}^r{a}_{\lambda }^p{a}_{\mu }^q),\end{array}$

where

$F=\frac{1}{2}{F}_{\lambda \mu }^{m}d{x}^{\lambda }\wedge d{x}^{\mu }\otimes {\mathrm{e}}_m$

is called the strength form of a principal connection.

See also

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

References

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