- Formal definition
- Properties
- See also
- Notes
- References
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.[1]
Formal definition
Let 
be a vector bundle with a typical fiber a vector space 
. An affine bundle modelled on a vector bundle is a fiber bundle 
whose typical fiber 
is an affine space modelled on so that the following conditions hold:
(i) All the fiber 
of 
are affine spaces modelled over the corresponding fibers 
of a vector bundle 
.
(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 
.
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, {{ISBN|0-471-15733-3}}.
- {{citation|last1 = Kolář|first1=Ivan|last2=Michor|first2=Peter|last3=Slovák|first3=Jan|url=http://www.emis.de/monographs/KSM/kmsbookh.pdf|format=PDF|title=Natural operators in differential geometry|year = 1993|publisher = Springer-Verlag}}
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, {{ISBN|978-3-659-37815-7}}; arXiv: 0908.1886.
- {{citation|last1 = Saunders|first1=D.J.|title=The geometry of jet bundles|year = 1989|publisher = Cambridge University Press|isbn=0-521-36948-7}}
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