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

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

(i) Every fiber ${\displaystyle Y_{x}}$ of ${\displaystyle Y}$ is an affine space modelled over the corresponding fibers ${\displaystyle {\overline {Y}}_{x}}$ of a vector bundle ${\displaystyle {\overline {Y}}}$.

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

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

There are the bundle morphisms

where ${\displaystyle ({\overline {y}}^{i})}$ are linear bundle coordinates on a vector bundle ${\displaystyle {\overline {Y}}}$, possessing linear transition functions ${\displaystyle {\overline {y}}'^{i}=A_{j}^{i}(x^{\nu }){\overline {y}}^{j}}$.

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let ${\displaystyle \pi :Y\to X}$ be an affine bundle modelled on a vector bundle ${\displaystyle {\overline {\pi }}:{\overline {Y}}\to X}$. Every global section ${\displaystyle s}$ of an affine bundle ${\displaystyle Y\to X}$ yields the bundle morphisms