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

(i) Every fiber [math]\displaystyle{ Y_x }[/math] of [math]\displaystyle{ Y }[/math] is an affine space modelled over the corresponding fibers [math]\displaystyle{ \overline Y_x }[/math] of a vector bundle [math]\displaystyle{ \overline Y }[/math].

(ii) There is an affine bundle atlas of [math]\displaystyle{ Y\to X }[/math] whose local trivializations morphisms and transition functions are affine isomorphisms.

Dealing with affine bundles, one uses only affine bundle coordinates [math]\displaystyle{ (x^\mu,y^i) }[/math] possessing affine transition functions

[math]\displaystyle{ y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu). }[/math]

There are the bundle morphisms

[math]\displaystyle{ Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, \overline y^i)\longmapsto y^i +\overline y^i, }[/math]

[math]\displaystyle{ Y\times_X Y\longrightarrow \overline Y,\qquad (y^i, y'^i)\longmapsto y^i - y'^i, }[/math]

where [math]\displaystyle{ (\overline y^i) }[/math] are linear bundle coordinates on a vector bundle [math]\displaystyle{ \overline Y }[/math], possessing linear transition functions [math]\displaystyle{ \overline y'^i= A^i_j(x^\nu)\overline y^j }[/math].

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 [math]\displaystyle{ \pi:Y\to X }[/math] be an affine bundle modelled on a vector bundle [math]\displaystyle{ \overline\pi:\overline Y\to X }[/math]. Every global section [math]\displaystyle{ s }[/math] of an affine bundle [math]\displaystyle{ Y\to X }[/math] yields the bundle morphisms

[math]\displaystyle{ 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. }[/math]

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

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

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

[math]\displaystyle{ \overline \Phi: \overline Y\to \overline Y', \qquad \overline y'^i= \frac{\partial\Phi^i}{\partial y^j}\overline y^j, }[/math]

called the linear derivative of [math]\displaystyle{ \Phi }[/math].

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