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Connection (affine bundle)

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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 → JY of the jet bundle JY → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the connection on an affine bundle is an example of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use of the adjective "affine".)

With respect to affine bundle coordinates (x, y) 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 ) . {\displaystyle {\begin{aligned}\Gamma &=dx^{\lambda }\otimes \left(\partial _{\lambda }+\Gamma _{\lambda }^{i}\partial _{i}\right)\,,\\\Gamma _{\lambda }^{i}&={{\Gamma _{\lambda }}^{i}}_{j}\left(x^{\nu }\right)y^{j}+\sigma _{\lambda }^{i}\left(x^{\nu }\right)\,.\end{aligned}}}

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 → JY, the corresponding linear derivative Γ : Y → JY of an affine morphism Γ defines a unique linear connection on a vector bundle Y → X. With respect to linear bundle coordinates (x, y) on Y, this connection reads

Γ ¯ = d x λ ⊗ ( ∂ λ + Γ λ i j ( x ν ) y ¯ j ∂ ¯ i ) . {\displaystyle {\overline {\Gamma }}=dx^{\lambda }\otimes \left(\partial _{\lambda }+{{\Gamma _{\lambda }}^{i}}_{j}\left(x^{\nu }\right){\overline {y}}^{j}{\overline {\partial }}_{i}\right)\,.}

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 . {\displaystyle \sigma =\sigma _{\lambda }^{i}(x^{\nu })dx^{\lambda }\otimes \partial _{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.

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 {\displaystyle \sigma =\sigma _{\lambda }^{i}(x^{\nu })dx^{\lambda }\otimes e_{i}}

where ei 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 , {\displaystyle {\begin{aligned}T&={\tfrac {1}{2}}T_{\lambda \mu }^{i}dx^{\lambda }\wedge dx^{\mu }\otimes \partial _{i}\,,\\T_{\lambda \mu }^{i}&=\partial _{\lambda }\sigma _{\mu }^{i}-\partial _{\mu }\sigma _{\lambda }^{i}+\sigma _{\lambda }^{h}{{\Gamma _{\mu }}^{i}}_{h}-\sigma _{\mu }^{h}{{\Gamma _{\lambda }}^{i}}_{h}\,,\end{aligned}}}

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

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

θ = d x μ ⊗ ∂ ˙ μ {\displaystyle \theta =dx^{\mu }\otimes {\dot {\partial }}_{\mu }}

on TX which coincides with the tautological one-form

θ X = d x μ ⊗ ∂ μ {\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu }}

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 , {\displaystyle {\begin{aligned}A&=\Gamma +\theta \,,\\A_{\lambda }^{\mu }&={{\Gamma _{\lambda }}^{\mu }}_{\nu }{\dot {x}}^{\nu }+\delta _{\lambda }^{\mu }\,,\end{aligned}}}

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.

See also

References

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