Connection (fibred manifold) - Alchetron, the free social encyclopedia

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:

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

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

H Y = Γ ( Y × X T X ) ⊂ 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

R ⊃ [ , ] ∋ t → x ( t ) ∈ X R ∋ t → y ( t ) ∈ Y

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

π ( y ( t ) ) = x ( t ) , y ˙ ( t ) ∈ H Y , t ∈ 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

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

θ X = d x μ ⊗ ∂ μ

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

Γ : ∂ λ → ∂ λ ⌋ Γ = ∂ λ + Γ λ i ∂ 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

Γ τ = τ ⌋ Γ = τ λ ( ∂ λ + Γ λ i ∂ i ) ⊂ 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 → Y × X T ∗ X → T ∗ Y → V ∗ Y → 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

Γ = ( d y i − Γ λ i d x λ ) ⊗ ∂ 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 ∗ Γ = ( d y i − ( Γ ∘ f ~ ) λ i ∂ f λ ∂ x ′ μ d x ′ μ ) ⊗ ∂ 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, yi
μ). Due to the canonical imbedding

J 1 Y → Y ( Y × X T ∗ X ) ⊗ Y T Y , ( y μ i ) → d x μ ⊗ ( ∂ μ + y μ i ∂ i ) ,

any connection Γ 3 on a fibered manifold Y → X is represented by a global section

Γ : Y → J 1 Y , y λ i ∘ Γ = Γ λ i ,

of the jet bundle J¹Y → Y, 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 Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential

R = 1 2 d Γ Γ = 1 2 [ Γ , Γ ] F N = 1 2 R λ μ i d x λ ∧ d x μ ⊗ ∂ i , R λ μ i = ∂ λ Γ μ i − ∂ μ Γ λ i + Γ λ j ∂ j Γ μ i − Γ μ j ∂ j Γ λ i .

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 Γ σ = ( ∂ λ σ μ i + Γ λ j ∂ j σ μ i − ∂ j Γ λ i σ μ j ) d x λ ∧ d x μ ⊗ ∂ 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, am
μ), and its sections are represented by vector-valued one-forms

A = d x λ ⊗ ( ∂ λ + a λ m e m ) ,

where

a λ m d x λ ⊗ 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

a λ μ r = 1 2 ( F λ μ r + S λ μ r ) = 1 2 ( a λ μ r + a μ λ r − c p q r a λ p a μ q ) + 1 2 ( a λ μ r − a μ λ r + c p q r a λ p a μ q ) ,

where

F = 1 2 F λ μ m d x λ ∧ d x μ ⊗ e m

is called the strength form of a principal connection.

References

Connection (fibred manifold) Wikipedia

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