Connection (affine bundle) explained

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

\begin{align}\Gamma & = dx ^{λ ⊗} \left(\partial _{λ} + \Gamma ^{i\partial} _{i\right),} \ \Gamma ^{\nu\right)} _{j\left(x} y ^{j} + \sigma ^{i\left(x} _{λ} ^{\nu\right).} \end{align}

An affine bundle is a fiber bundle with a general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affine connection is associated to a principal connection. It always exists. For any affine connection, the corresponding linear derivative of an affine morphism defines a unique linear connection on a vector bundle . With respect to linear bundle coordinates on, this connection reads

\overline \Gamma = dx ^{λ ⊗ \left(\partial} _{λ} + {{\Gamma ^{\nu\right)} _{j\left(x} \overline y ^{j\overline\partial} _{i\right).}

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

If is a vector bundle, both an affine connection and an associated linear connection areconnections on the same vector bundle, and their difference is a basic soldering form on

\sigma = \sigma ^{i(x} _{λ} ^{\nu)} dx ^{λ ⊗ \partial} _{i} . Thus, every affine connection on a vector bundle is a sum of a linear connection and a basic soldering form on .

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

\sigma = \sigma ^{i(x} _{λ} ^{\nu)} dx ^{λ ⊗} e _{i} where is a fiber basis for .

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

\begin{align} T & = \tfrac 12 T ^{i} _{λ \mu} dx ^{λ\wedge} dx ^{\mu ⊗} \partial _{i,} \\ T ^{i} _{λ \mu} & = \partial _{λ\sigma} ^{i} ^{\mu} - \partial _{\mu\sigma} ^{i} ^{λ} + \sigma ^{h} _{λ} {{\Gamma ^{i}} _{h} - \sigma ^{h} _{\mu} {{\Gamma ^{i}} _{h,} \end{align}

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

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

\theta = dx ^{\mu ⊗} • \partial _{\mu} on which coincides with the tautological one-form\theta ^{\mu ⊗} _{X = dx} \partial _{\mu} on due to the canonical vertical splitting . Given an arbitrary linear connection on, the corresponding affine connection\begin{align} A& = \Gamma + \theta, \\ A ^{\mu}} _{\nu} • x ^{\nu} + \delta ^{\mu} _{λ , \end{align}}

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

See also

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

References

• Book: S. . Kobayashi . K. . Nomizu . Foundations of Differential Geometry . 1–2 . Wiley-Interscience . 1996 . 0-471-15733-3.
• Book: Sardanashvily, G. . Gennadi Sardanashvily

. Gennadi Sardanashvily. Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory . Lambert Academic Publishing . 2013 . 978-3-659-37815-7 . 0908.1886. 2009arXiv0908.1886S .