Affine bundle

Short description: Type of fiber bundle

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

Formal definition

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

(i) Every fiber $Y_x$ of $Y$ is an affine space modelled over the corresponding fibers ${\bar{Y}}_x$ of a vector bundle $\bar{Y}$.

(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

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

There are the bundle morphisms

$Y{\times }_X\bar{Y}⟶Y,(y^i,\stackrel{i}{\stackrel{‾}{y}})⟼y^i+\stackrel{i}{\stackrel{‾}{y}},$

$Y{\times }_{X}Y⟶\bar{Y},(y^i,y{\text{'}}^i)⟼y^i-y{\text{'}}^i,$

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

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 $\bar{\pi }:\bar{Y}\to X$. Every global section $s$ of an affine bundle $Y\to X$ yields the bundle morphisms

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

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 $\mathrm{\Phi }:Y\to Y^{\prime }$ whose restriction to each fiber of $Y$ is an affine map. Every affine bundle morphism $\mathrm{\Phi }:Y\to Y^{\prime }$ of an affine bundle $Y$ modelled on a vector bundle $\bar{Y}$ to an affine bundle $Y^{\prime }$ modelled on a vector bundle ${\bar{Y}}^{\prime }$ yields a unique linear bundle morphism

$\bar{\mathrm{\Phi }}:\bar{Y}\to {\bar{Y}}^{\prime },\bar{y}{\text{'}}^i=\frac{\partial {\mathrm{\Phi }}^i}{\partial y^j}\stackrel{j}{\stackrel{‾}{y}},$

called the linear derivative of $\mathrm{\Phi }$.

See also

• Fiber bundle
• Fibered manifold
• Vector bundle
• Affine space

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) (PDF), Natural operators in differential geometry, Springer-Verlag, http://www.emis.de/monographs/KSM/kmsbookh.pdf, 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, https://archive.org/details/geometryofjetbun0000saun