Connection (fibred manifold)

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In differential geometry, a fibered manifold is surjective submersion of smooth manifolds YX. 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[edit]

Let π : YX be a fibered manifold. A generalized connection on Y is a section Γ : Y → J1Y, where J1Y is the jet manifold of Y.[1]

Connection as a horizontal splitting[edit]

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

 

 

 

 

(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 YX is defined as a linear bundle morphism

 

 

 

 

(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

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 YX 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

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

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 yπ−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[edit]

Given a fibered manifold YX, let it be endowed with an atlas of fibered coordinates (xμ, yi), and let Γ be a connection on YX. It yields uniquely the horizontal tangent-valued one-form

 

 

 

 

(3)

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

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

In particular, the connection Γ in 3 yields the horizontal lift of any vector field τ = τμμ on X to a projectable vector field

on Y.

Connection as a vertical-valued form[edit]

The horizontal splitting 2 of the exact sequence 1 defines the corresponding splitting of the dual exact sequence

where T*Y and T*X are the cotangent bundles of Y, respectively, and V*YY is the dual bundle to VYY, 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 YX, let f : X′ → X be a morphism and fYX the pullback bundle of Y by f. Then any connection Γ 3 on YX induces the pullback connection

on fYX.

Connection as a jet bundle section[edit]

Let J1Y be the jet manifold of sections of a fibered manifold YX, with coordinates (xμ, yi, yi
μ
)
. Due to the canonical imbedding

any connection Γ 3 on a fibered manifold YX is represented by a global section

of the jet bundle J1YY, and vice versa. It is an affine bundle modelled on a vector bundle

 

 

 

 

(4)

There are the following corollaries of this fact.

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

     

     

     

     

    (5)

    on YX, i.e., sections of the vector bundle 4.
  2. Connection coefficients possess the coordinate transformation law
  3. Every connection Γ on a fibred manifold YX yields the first order differential operator

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

    and the covariant derivative

    along a vector field τ on X are defined.

Curvature and torsion[edit]

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

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

Bundle of principal connections[edit]

Let π : PM 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 J1PP 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/GM, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle VP/GM 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 {em} for a Lie algebra of G, the fiber bundle C is endowed with bundle coordinates (xμ, am
μ
)
, and its sections are represented by vector-valued one-forms

where

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

where

is called the strength form of a principal connection.

See also[edit]

Notes[edit]

  1. ^ Krupka, Demeter; Janyška, Josef (1990). Lectures on differential invariants. Univerzita J. E. Purkyně v Brně. p. 174. ISBN 80-210-0165-8. 

References[edit]