In differential geometry, an area of mathematics, an affine bundle is a fiber bundle whose typical fiber is an affine space and whose trivialization morphisms and transition functions are affine.. (page 60)
Formal definition Let be a vector bundle with a typical fiber a vector space . An affine bundle modelled on is a fiber bundle whose typical fiber is an affine space modelled on so that the following conditions hold:
(i) Every fiber of is an affine space modelled over the corresponding fiber of the vector bundle .
(ii) There is a bundle atlas of whose local trivialization morphisms and transition functions are affine isomorphisms. Such an atlas is called an affine bundle atlas.
When dealing with affine bundles, one uses 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 .
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 be an affine bundle modelled on a vector bundle . Every global section 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 of a manifold 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 of dimension . 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 whose restriction to each fiber of is an affine map. Every affine bundle morphism of an affine bundle modelled on a vector bundle to an affine bundle modelled on a vector bundle yields a unique linear bundle morphism
called the linear derivative of .
See also Fiber bundle Fibered manifold Vector bundle Affine space
Notes
References S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, 1996, .
Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, ; .