Affine bundle
# 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

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

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 $\Phi$.

