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

[math]\displaystyle{ \begin{align}\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{align} }[/math]

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

[math]\displaystyle{ \overline \Gamma=dx^\lambda\otimes\left(\partial_\lambda +{{\Gamma_\lambda}^i}_j\left(x^\nu\right) \overline y^j\overline\partial_i\right)\,. }[/math]

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

[math]\displaystyle{ \sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,. }[/math]

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

[math]\displaystyle{ \sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i }[/math]

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

[math]\displaystyle{ \begin{align} T &=\tfrac12 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{align} }[/math]

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

[math]\displaystyle{ \theta=dx^\mu\otimes \dot\partial_\mu }[/math]

on TX which coincides with the tautological one-form

[math]\displaystyle{ \theta_X=dx^\mu\otimes \partial_\mu }[/math]

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

[math]\displaystyle{ \begin{align} A&=\Gamma +\theta\,, \\ A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,, \end{align} }[/math]

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

• Connection (fibred manifold)
• Affine connection
• Connection (vector bundle)
• Connection (mathematics)
• Affine gauge theory

References

• Kobayashi, S.; Nomizu, K. (1996). Foundations of Differential Geometry. 1–2. Wiley-Interscience. ISBN 0-471-15733-3. 
• Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory. Lambert Academic Publishing. ISBN 978-3-659-37815-7. Bibcode: 2009arXiv0908.1886S.