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