Connection (affine bundle)

In differential geometry, a connection on an affine bundle is a specialisation to affine bundles of the more general notion of a connection on a principal bundle. 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 an affine connection if, as a section $Γ : Y \to J 1 Y$ of the jet bundle $J 1 Y \to Y$ of $Y$ , it is an affine bundle morphism over $X$ .

The term "affine connection" as used in this article should not be confused with its more common usage, namely a connection on the tangent bundle $T X$ of a smooth manifold $X$ , though as discussed below, the latter can be considered as a special example of the former.

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{matrix} Γ & = d x^{λ} \otimes (\partial_{λ} + Γ_{λ}^{i} \partial_{i}), \\ Γ_{λ}^{i} & = {Γ_{λ}}^{i}_{j} (x^{ν}) y^{j} + σ_{λ}^{i} (x^{ν}) . \end{matrix}

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{Γ} = d x^{λ} \otimes (\partial_{λ} + {Γ_{λ}}^{i}_{j} (x^{ν}) \overset{j}{\overline{y}} {\overline{\partial}}_{i}) .

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

σ = σ_{λ}^{i} (x^{ν}) d x^{λ} \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

σ = σ_{λ}^{i} (x^{ν}) d x^{λ} \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{matrix} T & = \frac{1}{2} T_{λ μ}^{i} d x^{λ} \land d x^{μ} \otimes \partial_{i}, \\ T_{λ μ}^{i} & = \partial_{λ} σ_{μ}^{i} - \partial_{μ} σ_{λ}^{i} + σ_{λ}^{h} {Γ_{μ}}^{i}_{h} - σ_{μ}^{h} {Γ_{λ}}^{i}_{h}, \end{matrix}

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

θ = d x^{μ} \otimes {\dot{\partial}}_{μ}

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

θ_{X} = d x^{μ} \otimes \partial_{μ}

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{matrix} A & = Γ + θ, \\ A_{λ}^{μ} & = {Γ_{λ}}^{μ}_{ν} {\dot{x}}^{ν} + δ_{λ}^{μ}, \end{matrix}

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

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. Search this book on

Template:Differential-geometry-stub

This article "Connection (affine bundle)" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Connection (affine bundle). Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

Connection (affine bundle)

See also

References

📰 Article(s) of the same category(ies)