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 formula_1 be a vector bundle with a typical fiber a vector space formula_2. An affine bundle modelled on a vector bundle formula_1 is a fiber bundle formula_4 whose typical fiber formula_5 is an affine space modelled on formula_2 so that the following conditions hold:
(i) All the fiber formula_7 of formula_8 are affine spaces modelled over the corresponding fibers formula_9 of a vector bundle formula_10.
(ii) There is an affine bundle atlas of formula_11 whose local trivializations morphisms and transition functions are affine isomorphisms.
Dealing with affine bundles, one uses only affine bundle coordinates formula_12 possessing affine transition functions
There are the bundle morphisms
where formula_16 are linear bundle coordinates on a vector bundle formula_10, possessing linear transition functions formula_18.
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 formula_4 be an affine bundle modelled on a vector bundle formula_1. Every global section formula_21 of an affine bundle formula_11 yields the bundle morphisms
In particular, every vector bundle formula_8 has a natural structure of an affine bundle due to these morphisms where formula_25 is the canonical zero-valued section of formula_8. For instance, the tangent bundle formula_27 of a manifold formula_28 naturally is an affine bundle. 
An affine bundle formula_11 is a fiber bundle with a general affine structure group formula_30 of affine transformations of its typical fiber formula_31 of dimension formula_32. This structure group always is reducible to a general linear group formula_33, i.e., an affine bundle admits an atlas with linear transition functions.
By a morphism of affine bundles is meant a bundle morphism formula_34 whose restriction to each fiber of formula_8 is an affine map. Every affine bundle morphism formula_34 of an affine bundle formula_8 modelled on a vector bundle formula_10 to an affine bundle formula_39 modelled on a vector bundle formula_40 yields a unique linear bundle morphism
called the "linear derivative" of formula_42.