Affine bundle

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) All the fiber $Y_{x}$ of $Y$ are affine spaces 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 $\Phi$.

See also

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

Notes

1. Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag. (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
• 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