Affine bundle

In differential geometry, an area of mathematics, an affine bundle is a fiber bundle whose typical fiber is an affine space and whose trivialization morphisms and transition functions are affine.

Formal definition

Let ${\overline {\pi }}:{\overline {Y}}\to X$ be a vector bundle with a typical fiber a vector space ${\overline {F}}$ . An affine bundle modelled on ${\overline {\pi }}:{\overline {Y}}\to X$ is a fiber bundle $\pi :Y\to X$ whose typical fiber $F$ is an affine space modelled on ${\overline {F}}$ so that the following conditions hold:

(i) Every fiber $Y_{x}$ of $Y$ is an affine space modelled over the corresponding fiber ${\overline {Y}}_{x}$ of the vector bundle ${\overline {Y}}$ .

(ii) There is a bundle atlas of $Y\to X$ whose local trivialization morphisms and transition functions are affine isomorphisms. Such an atlas is called an affine bundle atlas.

When dealing with affine bundles, one uses affine bundle coordinates $(x^{\mu },y^{i})$ possessing affine transition functions

y'^{i}=A_{j}^{i}(x^{\nu })y^{j}+b^{i}(x^{\nu }).

There are the bundle morphisms

Y\times _{X}{\overline {Y}}\longrightarrow Y,\qquad (y^{i},{\overline {y}}^{i})\longmapsto y^{i}+{\overline {y}}^{i},

Y\times _{X}Y\longrightarrow {\overline {Y}},\qquad (y^{i},y'^{i})\longmapsto y^{i}-y'^{i},

where $({\overline {y}}^{i})$ are linear bundle coordinates on a vector bundle ${\overline {Y}}$ , possessing linear transition functions ${\overline {y}}'^{i}=A_{j}^{i}(x^{\nu }){\overline {y}}^{j}$ .

Properties

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let $\pi :Y\to X$ be an affine bundle modelled on a vector bundle ${\overline {\pi }}:{\overline {Y}}\to X$ . Every global section $s$ of an affine bundle $Y\to X$ yields the bundle morphisms

Y\ni y\to y-s(\pi (y))\in {\overline {Y}},\qquad {\overline {Y}}\ni {\overline {y}}\to s(\pi (y))+{\overline {y}}\in Y.

In particular, every vector bundle $Y$ has a natural structure of an affine bundle due to these morphisms where $s=0$ is the canonical zero-valued section of $Y$ . For instance, the tangent bundle $TX$ of a manifold $X$ naturally is an affine bundle.

An affine bundle $Y\to X$ is a fiber bundle with a general affine structure group $GA(m,\mathbb {R} )$ of affine transformations of its typical fiber $V$ of dimension $m$ . This structure group always is reducible to a general linear group $GL(m,\mathbb {R} )$ , i.e., an affine bundle admits an atlas with linear transition functions.

By a morphism of affine bundles is meant a bundle morphism $\Phi :Y\to Y'$ whose restriction to each fiber of $Y$ is an affine map. Every affine bundle morphism $\Phi :Y\to Y'$ of an affine bundle $Y$ modelled on a vector bundle ${\overline {Y}}$ to an affine bundle $Y'$ modelled on a vector bundle ${\overline {Y}}'$ yields a unique linear bundle morphism

{\overline {\Phi }}:{\overline {Y}}\to {\overline {Y}}',\qquad {\overline {y}}'^{i}={\frac {\partial \Phi ^{i}}{\partial y^{j}}}{\overline {y}}^{j},

called the linear derivative of $\Phi$ .

Connection

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 $X$ . This is closely related to the general meaning of an affine connection on the tangent bundle $T X$ of a smooth manifold $X$ .

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 $Γ$ .

Notes

↑ Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2013-05-28. (page 60)

References

S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, 1996, ISBN 0-471-15733-3.
Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2013-05-28
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.
Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7

[1] Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2013-05-28. (page 60)