Connection (affine bundle)

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 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^λ, yⁱ) on Y, an affine connection Γ on Y → X is given by the tangent-valued connection form

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

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 → 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

${\displaystyle {\overline {\Gamma }}=dx^{\lambda }\otimes \left(\partial _{\lambda }+{{\Gamma _{\lambda }}^{i}}_{j}\left(x^{\nu }\right){\overline {y}}^{j}{\overline {\partial }}_{i}\right)\,.}$

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

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

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

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

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

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

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^μ, ẋ^μ). There is the canonical soldering form

${\displaystyle \theta =dx^{\mu }\otimes {\dot {\partial }}_{\mu }}$

on TX which coincides with the tautological one-form

${\displaystyle \theta _{X}=dx^{\mu }\otimes \partial _{\mu }}$

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

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

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.