Affine bundle

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, 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 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


 * $$y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu).$$

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 $$\overline y'^i= A^i_j(x^\nu)\overline y^j $$.

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 $$\Phi$$.