Connection (affine bundle) - Alchetron, the free social encyclopedia

Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J¹Y of the jet bundle J¹Y → Y of Y is an affine bundle morphism over X. In particular, this is the case of an affine connection on the tangent bundle TX of a smooth manifold X.

With respect to affine bundle coordinates (x^λ, yⁱ) on Y, an affine connection Γ on Y → X is given by the tangent-valued connection form

Γ = d x λ ⊗ ( ∂ λ + Γ λ i ∂ i ) , Γ λ i = Γ λ i j ( x ν ) y j + σ λ i ( x ν ) .

An affine bundle is a fiber bundle with a general affine structure group GA(m, ℝ) of affine transformations of its typical fiber V of dimension m. Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection Γ : Y → J¹Y, the corresponding linear derivative Γ : Y → J¹Y of an affine morphism Γ defines a unique linear connection on a vector bundle Y → X. With respect to linear bundle coordinates (x^λ, yⁱ) on Y, this connection reads

Γ ¯ = d x λ ⊗ ( ∂ λ + Γ λ i j ( x ν ) y ¯ j ∂ ¯ i ) .

Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.

If Y → X is a vector bundle, both an affine connection Γ and an associated linear connection Γ are connections on the same vector bundle Y → X, and their difference is a basic soldering form on

σ = σ λ i ( x ν ) d x λ ⊗ ∂ i .

Thus, every affine connection on a vector bundle Y → X is a sum of a linear connection and a basic soldering form on Y → X.

It should be noted that, due to the canonical vertical splitting VY = Y × Y, this soldering form is brought into a vector-valued form

σ = σ λ i ( x ν ) d x λ ⊗ e i

where e_i is a fiber basis for Y.

Given an affine connection Γ on a vector bundle Y → X, let R and R be the curvatures of a connection Γ and the associated linear connection Γ, respectively. It is readily observed that R = R + T, where

T = 1 2 T λ μ i d x λ ∧ d x μ ⊗ ∂ i , T λ μ i = ∂ λ σ μ i − ∂ μ σ λ i + σ λ h Γ μ i h − σ μ h Γ λ i h ,

is the torsion of Γ with respect to the basic soldering form σ.

In particular, let us consider the tangent bundle TX of a manifold X coordinated by (x^μ, ẋ^μ). There is the canonical soldering form

θ = d x μ ⊗ ∂ ˙ μ

on TX which coincides with the tautological one-form

θ X = d x μ ⊗ ∂ μ

on X due to the canonical vertical splitting VTX = TX × TX. Given an arbitrary linear connection Γ on TX, the corresponding affine connection

A = Γ + θ , A λ μ = Γ λ μ ν x ˙ ν + δ λ μ ,

on TX is the Cartan connection. The torsion of the Cartan connection A with respect to the soldering form θ coincides with the torsion of a linear connection Γ, and its curvature is a sum R + T of the curvature and the torsion of Γ.

References

Connection (affine bundle) Wikipedia

(Text) CC BY-SA