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Connection (affine bundle)

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Let $Y \to X$ be an affine bundle modelled over a vector bundle $Y \to X$ . A connection $Γ$ on $Y \to X$ is called the affine connection if it as a section $Γ : Y \to J 1 Y$ of the jet bundle $J 1 Y \to Y$ of $Y$ is an affine bundle morphism over $X$ . In particular, this is an affine connection on the tangent bundle $T X$ 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 $(x λ, y i)$ on $Y$ , an affine connection $Γ$ on $Y \to X$ is given by the tangent-valued connection form

{\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 $Γ : Y \to J 1 Y$ , the corresponding linear derivative $Γ : Y \to J 1 Y$ of an affine morphism $Γ$ defines a unique linear connection on a vector bundle $Y \to X$ . With respect to linear bundle coordinates $(x λ, y i)$ on $Y$ , this connection reads

{\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 $Y \to X$ is a vector bundle, both an affine connection $Γ$ and an associated linear connection $Γ$ are connections on the same vector bundle $Y \to X$ , and their difference is a basic soldering form on

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

Thus, every affine connection on a vector bundle $Y \to X$ is a sum of a linear connection and a basic soldering form on $Y \to X$ .

Due to the canonical vertical splitting $V Y = Y \times Y$ , this soldering form is brought into a vector-valued form

\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 $Y \to X$ , 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

{\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 $T X$ of a manifold $X$ coordinated by $(x μ, ẋ μ)$ . There is the canonical soldering form

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

on $T X$ which coincides with the tautological one-form

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

on $X$ due to the canonical vertical splitting $VT X = T X \times T X$ . Given an arbitrary linear connection $Γ$ on $T X$ , the corresponding affine connection

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

on $T X$ 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 $Γ$ .

References

Kobayashi, S.; Nomizu, K. (1996). Foundations of Differential Geometry. Vol. 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. arXiv:0908.1886. Bibcode:2009arXiv0908.1886S. ISBN 978-3-659-37815-7.

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References