Affine bundle

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

be a vector bundle with a typical fiber a vector space

An affine bundle modelled on a vector bundle

is an affine space modelled on

so that the following conditions hold: (i) Every fiber

is an affine space modelled over the corresponding fibers

(ii) There is an affine bundle atlas of

whose local trivializations morphisms and transition functions are affine isomorphisms.

Dealing with affine bundles, one uses only affine bundle coordinates

possessing affine transition functions There are the bundle morphisms where

are linear bundle coordinates on a vector bundle

, possessing linear transition functions

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle.

be an affine bundle modelled on a vector bundle

of an affine bundle

yields the bundle morphisms In particular, every vector bundle

has a natural structure of an affine bundle due to these morphisms where

is the canonical zero-valued section of

For instance, the tangent bundle

naturally is an affine bundle.

An affine bundle

is a fiber bundle with a general affine structure group

of affine transformations of its typical fiber

This structure group always is reducible to a general linear group

, i.e., an affine bundle admits an atlas with linear transition functions.

By a morphism of affine bundles is meant a bundle morphism

Every affine bundle morphism

modelled on a vector bundle

modelled on a vector bundle

yields a unique linear bundle morphism called the linear derivative of