Affine bundle

Last updated November 03, 2021

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.^[1]

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 a vector bundle ${\overline {\pi }}:{\overline {Y}}\to X$ 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 fibers ${\overline {Y}}_{x}$ of a vector bundle ${\overline {Y}}$ .

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

Dealing with affine bundles, one uses only 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 $\Phi$ .

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)

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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., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; arXiv : 0908.1886.
Saunders, D.J. (1989), The geometry of jet bundles , Cambridge University Press, ISBN 0-521-36948-7

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[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)

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