Английская Википедия:Connection (affine bundle)
Let Шаблон:Math be an affine bundle modelled over a vector bundle Шаблон:Math. A connection Шаблон:Math on Шаблон:Math is called the affine connection if it as a section Шаблон:Math of the jet bundle Шаблон:Math of Шаблон:Math is an affine bundle morphism over Шаблон:Math. In particular, this is an affine connection on the tangent bundle Шаблон:Math of a smooth manifold Шаблон:Math. (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 Шаблон:Math on Шаблон:Math, an affine connection Шаблон:Math on Шаблон:Math is given by the tangent-valued connection form
- <math>\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 Шаблон:Math of affine transformations of its typical fiber Шаблон:Math of dimension Шаблон:Math. Therefore, an affine connection is associated to a principal connection. It always exists.
For any affine connection Шаблон:Math, the corresponding linear derivative Шаблон:Math of an affine morphism Шаблон:Math defines a unique linear connection on a vector bundle Шаблон:Math. With respect to linear bundle coordinates Шаблон:Math on Шаблон:Math, this connection reads
- <math> \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 Шаблон:Math is a vector bundle, both an affine connection Шаблон:Math and an associated linear connection Шаблон:Math are connections on the same vector bundle Шаблон:Math, and their difference is a basic soldering form on
- <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes\partial_i \,.</math>
Thus, every affine connection on a vector bundle Шаблон:Math is a sum of a linear connection and a basic soldering form on Шаблон:Math.
Due to the canonical vertical splitting Шаблон:Math, this soldering form is brought into a vector-valued form
- <math>\sigma= \sigma_\lambda^i(x^\nu) dx^\lambda\otimes e_i </math>
where Шаблон:Math is a fiber basis for Шаблон:Math.
Given an affine connection Шаблон:Math on a vector bundle Шаблон:Math, let Шаблон:Math and Шаблон:Math be the curvatures of a connection Шаблон:Math and the associated linear connection Шаблон:Math, respectively. It is readily observed that Шаблон:Math, where
- <math>\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 Шаблон:Math with respect to the basic soldering form Шаблон:Math.
In particular, consider the tangent bundle Шаблон:Math of a manifold Шаблон:Math coordinated by Шаблон:Math. There is the canonical soldering form
- <math>\theta=dx^\mu\otimes \dot\partial_\mu </math>
on Шаблон:Math which coincides with the tautological one-form
- <math>\theta_X=dx^\mu\otimes \partial_\mu</math>
on Шаблон:Math due to the canonical vertical splitting Шаблон:Math. Given an arbitrary linear connection Шаблон:Math on Шаблон:Math, the corresponding affine connection
- <math>\begin{align}
A&=\Gamma +\theta\,, \\ A_\lambda^\mu&={{\Gamma_\lambda}^\mu}_\nu \dot x^\nu +\delta^\mu_\lambda\,, \end{align}</math>
on Шаблон:Math is the Cartan connection. The torsion of the Cartan connection Шаблон:Math with respect to the soldering form Шаблон:Math coincides with the torsion of a linear connection Шаблон:Math, and its curvature is a sum Шаблон:Math of the curvature and the torsion of Шаблон:Math.
See also
- Connection (fibred manifold)
- Affine connection
- Connection (vector bundle)
- Connection (mathematics)
- Affine gauge theory
References
Шаблон:Differential-geometry-stub
