Connection (affine bundle)

Let Template:Math be an affine bundle modelled over a vector bundle Template:Math. A connection Template:Math on Template:Math is called the affine connection if it as a section Template:Math of the jet bundle Template:Math of Template:Math is an affine bundle morphism over Template:Math. In particular, this is an affine connection on the tangent bundle Template:Math of a smooth manifold Template: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 Template:Math on Template:Math, an affine connection Template:Math on Template:Math is given by the tangent-valued connection form

$\begin{array}{cc}\mathrm{\Gamma }& =d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{\mathrm{\Gamma }}_{\lambda }^i{\partial }_i),\\ {\mathrm{\Gamma }}_{\lambda }^i& ={{{\mathrm{\Gamma }}_{\lambda }}^i}_j\left(x^{\nu }\right)y^j+{\sigma }_{\lambda }^i\left(x^{\nu }\right).\end{array}$

An affine bundle is a fiber bundle with a general affine structure group Template:Math of affine transformations of its typical fiber Template:Math of dimension Template:Math. Therefore, an affine connection is associated to a principal connection. It always exists.

For any affine connection Template:Math, the corresponding linear derivative Template:Math of an affine morphism Template:Math defines a unique linear connection on a vector bundle Template:Math. With respect to linear bundle coordinates Template:Math on Template:Math, this connection reads

$\bar{\mathrm{\Gamma }}=d{x}^{\lambda }\otimes ({\partial }_{\lambda }+{{{\mathrm{\Gamma }}_{\lambda }}^i}_j\left(x^{\nu }\right)\stackrel{j}{\stackrel{‾}{y}}{\bar{\partial }}_i).$

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

If Template:Math is a vector bundle, both an affine connection Template:Math and an associated linear connection Template:Math are connections on the same vector bundle Template:Math, and their difference is a basic soldering form on

$\sigma ={\sigma }_{\lambda }^i(x^{\nu })d{x}^{\lambda }\otimes {\partial }_i.$

Thus, every affine connection on a vector bundle Template:Math is a sum of a linear connection and a basic soldering form on Template:Math.

Due to the canonical vertical splitting Template:Math, this soldering form is brought into a vector-valued form

$\sigma ={\sigma }_{\lambda }^i(x^{\nu })d{x}^{\lambda }\otimes e_i$

where Template:Math is a fiber basis for Template:Math.

Given an affine connection Template:Math on a vector bundle Template:Math, let Template:Math and Template:Math be the curvatures of a connection Template:Math and the associated linear connection Template:Math, respectively. It is readily observed that Template:Math, where

$\begin{array}{cc}T& =\frac{1}{2}{T}_{\lambda \mu }^{i}d{x}^{\lambda }\wedge d{x}^{\mu }\otimes {\partial }_i,\\ T_{\lambda \mu }^i& ={\partial }_{\lambda }{\sigma }_{\mu }^i-{\partial }_{\mu }{\sigma }_{\lambda }^i+{\sigma }_{\lambda }^h{{{\mathrm{\Gamma }}_{\mu }}^i}_h-{\sigma }_{\mu }^h{{{\mathrm{\Gamma }}_{\lambda }}^i}_h,\end{array}$

is the torsion of Template:Math with respect to the basic soldering form Template:Math.

In particular, consider the tangent bundle Template:Math of a manifold Template:Math coordinated by Template:Math. There is the canonical soldering form

$\theta =d{x}^{\mu }\otimes {\dot{\partial }}_{\mu }$

on Template:Math which coincides with the tautological one-form

${\theta }_X=d{x}^{\mu }\otimes {\partial }_{\mu }$

on Template:Math due to the canonical vertical splitting Template:Math. Given an arbitrary linear connection Template:Math on Template:Math, the corresponding affine connection

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

on Template:Math is the Cartan connection. The torsion of the Cartan connection Template:Math with respect to the soldering form Template:Math coincides with the torsion of a linear connection Template:Math, and its curvature is a sum Template:Math of the curvature and the torsion of Template:Math.

See also

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

References

• Template:Cite book
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