In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.[1]
Let π π --> ¯ ¯ --> : Y ¯ ¯ --> → → --> X {\displaystyle {\overline {\pi }}:{\overline {Y}}\to X} be a vector bundle with a typical fiber a vector space F ¯ ¯ --> {\displaystyle {\overline {F}}} . An affine bundle modelled on a vector bundle π π --> ¯ ¯ --> : Y ¯ ¯ --> → → --> X {\displaystyle {\overline {\pi }}:{\overline {Y}}\to X} is a fiber bundle π π --> : Y → → --> X {\displaystyle \pi :Y\to X} whose typical fiber F {\displaystyle F} is an affine space modelled on F ¯ ¯ --> {\displaystyle {\overline {F}}} so that the following conditions hold:
(i) Every fiber Y x {\displaystyle Y_{x}} of Y {\displaystyle Y} is an affine space modelled over the corresponding fibers Y ¯ ¯ --> x {\displaystyle {\overline {Y}}_{x}} of a vector bundle Y ¯ ¯ --> {\displaystyle {\overline {Y}}} .
(ii) There is an affine bundle atlas of Y → → --> X {\displaystyle Y\to X} whose local trivializations morphisms and transition functions are affine isomorphisms.
Dealing with affine bundles, one uses only affine bundle coordinates ( x μ μ --> , y i ) {\displaystyle (x^{\mu },y^{i})} possessing affine transition functions
There are the bundle morphisms
where ( y ¯ ¯ --> i ) {\displaystyle ({\overline {y}}^{i})} are linear bundle coordinates on a vector bundle Y ¯ ¯ --> {\displaystyle {\overline {Y}}} , possessing linear transition functions y ¯ ¯ --> ′ i = A j i ( x ν ν --> ) y ¯ ¯ --> j {\displaystyle {\overline {y}}'^{i}=A_{j}^{i}(x^{\nu }){\overline {y}}^{j}} .
An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let π π --> : Y → → --> X {\displaystyle \pi :Y\to X} be an affine bundle modelled on a vector bundle π π --> ¯ ¯ --> : Y ¯ ¯ --> → → --> X {\displaystyle {\overline {\pi }}:{\overline {Y}}\to X} . Every global section s {\displaystyle s} of an affine bundle Y → → --> X {\displaystyle Y\to X} yields the bundle morphisms
In particular, every vector bundle Y {\displaystyle Y} has a natural structure of an affine bundle due to these morphisms where s = 0 {\displaystyle s=0} is the canonical zero-valued section of Y {\displaystyle Y} . For instance, the tangent bundle T X {\displaystyle TX} of a manifold X {\displaystyle X} naturally is an affine bundle.
An affine bundle Y → → --> X {\displaystyle Y\to X} is a fiber bundle with a general affine structure group G A ( m , R ) {\displaystyle GA(m,\mathbb {R} )} of affine transformations of its typical fiber V {\displaystyle V} of dimension m {\displaystyle m} . This structure group always is reducible to a general linear group G L ( m , R ) {\displaystyle GL(m,\mathbb {R} )} , i.e., an affine bundle admits an atlas with linear transition functions.
By a morphism of affine bundles is meant a bundle morphism Φ Φ --> : Y → → --> Y ′ {\displaystyle \Phi :Y\to Y'} whose restriction to each fiber of Y {\displaystyle Y} is an affine map. Every affine bundle morphism Φ Φ --> : Y → → --> Y ′ {\displaystyle \Phi :Y\to Y'} of an affine bundle Y {\displaystyle Y} modelled on a vector bundle Y ¯ ¯ --> {\displaystyle {\overline {Y}}} to an affine bundle Y ′ {\displaystyle Y'} modelled on a vector bundle Y ¯ ¯ --> ′ {\displaystyle {\overline {Y}}'} yields a unique linear bundle morphism
called the linear derivative of Φ Φ --> {\displaystyle \Phi } .
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