Connection (fibred manifold)

Short description: Operation on fibered manifolds

In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Y → X. 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 π : Y → X be a fibered manifold. A generalized connection on Y is a section Γ : Y → J¹Y, where J¹Y is the jet manifold of Y.

Connection as a horizontal splitting

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

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

(1)

where TY and TX are the tangent bundles of Y, respectively, VY is the vertical tangent bundle of Y, and Y ×_X TX is the pullback bundle of TX onto Y.

A connection on a fibered manifold Y → X is defined as a linear bundle morphism

$\mathrm{\Gamma }:Y{\times }_X\mathrm{T}X\to \mathrm{T}Y$

(2)

over Y which splits the exact sequence 1. A connection always exists.

Sometimes, this connection Γ 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 TY and its horizontal decomposition TY = VY ⊕ HY.

At the same time, by an Ehresmann connection also is meant the following construction. Any connection Γ on a fibered manifold Y → X yields a horizontal lift Γ ∘ τ of a vector field τ on X onto Y, but need not defines the similar lift of a path in X into Y. Let

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

be two smooth paths in X and Y, respectively. Then t → y(t) is called the horizontal lift of x(t) if

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

A connection Γ is said to be the Ehresmann connection if, for each path x([0,1]) in X, there exists its horizontal lift through any point y ∈ π⁻¹(x([0,1])). 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 Y → X, let it be endowed with an atlas of fibered coordinates (x^μ, yⁱ), and let Γ be a connection on Y → X. It yields uniquely the horizontal tangent-valued one-form

$\mathrm{\Gamma }=d{x}^{\mu }\otimes ({\partial }_{\mu }+{\mathrm{\Gamma }}_{\mu }^i(x^{\nu },y^j){\partial }_i)$

(3)

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

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

on X, and vice versa. With this form, the horizontal splitting 2 reads

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

In particular, the connection Γ in 3 yields the horizontal lift of any vector field τ = τ^μ ∂_μ on X 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 Y.

Connection as a vertical-valued form

The horizontal splitting 2 of the exact sequence 1 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 T*Y and T*X are the cotangent bundles of Y, respectively, and V*Y → Y is the dual bundle to VY → Y, 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 Y → X, let f : X′ → X be a morphism and f ∗ Y → X′ the pullback bundle of Y by f. Then any connection Γ 3 on Y → X 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 f ∗ Y → X′.

Connection as a jet bundle section

Let J¹Y be the jet manifold of sections of a fibered manifold Y → X, with coordinates (x^μ, yⁱ, yiμ). 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 Γ 3 on a fibered manifold Y → X 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 J¹Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle

$\left(Y{\times }_X{T}^{*}X\right){\otimes }_Y\mathrm{V}Y\to Y.$

(4)

There are the following corollaries of this fact.

1. Connections on a fibered manifold Y → X make up an affine space modelled on the vector space of soldering forms

   $\sigma ={\sigma }_{\mu }^{i}d{x}^{\mu }\otimes {\partial }_i$

   (5)

   on Y → X, i.e., sections of the vector bundle 4.
2. Connection coefficients possess the coordinate transformation law

   ${{\mathrm{\Gamma }}^{\prime }}_{\lambda }^i=\frac{\partial x^{\mu }}{\partial {x^{\prime }}^{\lambda }}({\partial }_{\mu }{y^{\prime }}^i+{\mathrm{\Gamma }}_{\mu }^j{\partial }_j{y^{\prime }}^i).$

3. Every connection Γ on a fibred manifold Y → X yields the first order differential operator

   $D_{\mathrm{\Gamma }}:{\mathrm{J}}^{1}Y{\to }_Y{\mathrm{T}}^{*}X{\otimes }_Y\mathrm{V}Y,D_{\mathrm{\Gamma }}=(y_{\lambda }^i-{\mathrm{\Gamma }}_{\lambda }^i)d{x}^{\lambda }\otimes {\partial }_i,$

   on Y called the covariant differential relative to the connection Γ. If s : X → Y is a section, its covariant differential

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

   and the covariant derivative

   $\underset{\tau }{\overset{\mathrm{\Gamma }}{\mathrm{\nabla }}}s=\tau \rfloor {\mathrm{\nabla }}^{\mathrm{\Gamma }}s$

   along a vector field τ on X are defined.

Curvature and torsion

Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential

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

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

Given the connection Γ 3 and the soldering form σ 5, a torsion of Γ with respect to σ 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 π : P → M be a principal bundle with a structure Lie group G. A principal connection on P usually is described by a Lie algebra-valued connection one-form on P. At the same time, a principal connection on P is a global section of the jet bundle J¹P → P which is equivariant with respect to the canonical right action of G in P. Therefore, it is represented by a global section of the quotient bundle C = J¹P/G → M, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle VP/G → M whose typical fiber is the Lie algebra g of structure group G, and where G acts on by the adjoint representation. There is the canonical imbedding of C to the quotient bundle TP/G which also is called the bundle of principal connections.

Given a basis {e_m} for a Lie algebra of G, the fiber bundle C is endowed with bundle coordinates (x^μ, amμ), 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 M.

Let us note that the jet bundle J¹C of C is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition

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

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

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993). Natural operators in differential geometry. Springer-Verlag. http://www.emis.de/monographs/KSM/kmsbookh.pdf. Retrieved 2013-05-28. 
• Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants. Univerzita J. E. Purkyně v Brně. ISBN 80-210-0165-8. 
• Saunders, D.J. (1989). The geometry of jet bundles. Cambridge University Press. ISBN 0-521-36948-7. https://archive.org/details/geometryofjetbun0000saun. 
• Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory. World Scientific. ISBN 981-02-2013-8. 
• Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory. Lambert Academic Publishing. ISBN 978-3-659-37815-7. Bibcode: 2009arXiv0908.1886S.