Affine bundle

 ( Differential Geometry ) 

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60)

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Formal definition Let $\backslash overline\backslash pi:\backslash overline\; Y\backslash to\; X$ be a vector bundle with a typical fiber a vector space $\backslash overline\; F$. An affine bundle modelled on a vector bundle $\backslash overline\backslash pi:\backslash overline\; Y\backslash to\; X$ is a fiber bundle $\backslash pi:Y\backslash to\; X$ whose typical fiber $F$ is an affine space modelled on $\backslash overline\; F$ so that the following conditions hold:

(i) Every fiber $Y\_x$ of $Y$ is an affine space modelled over the corresponding fibers $\backslash overline\; Y\_x$ of a vector bundle $\backslash overline\; Y$.

(ii) There is an affine bundle atlas of $Y\backslash to\; X$ whose local trivializations morphisms and transition functions are affine isomorphisms.

Dealing with affine bundles, one uses only affine bundle coordinates $(x^\backslash mu,y^i)$ possessing affine transition functions

$y\text{'}^i=\; A^i\_j(x^\backslash nu)y^j\; +\; b^i(x^\backslash nu).$

There are the Bundle map

$Y\backslash times\_X\backslash overline\; Y\backslash longrightarrow\; Y,\backslash qquad\; (y^i,\; \backslash overline\; y^i)\backslash longmapsto\; y^i\; +\backslash overline\; y^i,$

$Y\backslash times\_X\; Y\backslash longrightarrow\; \backslash overline\; Y,\backslash qquad\; (y^i,\; y\text{'}^i)\backslash longmapsto\; y^i\; -\; y\text{'}^i,$

where $(\backslash overline\; y^i)$ are linear bundle coordinates on a vector bundle $\backslash overline\; Y$, possessing linear transition functions $\backslash overline\; y\text{'}^i=\; A^i\_j(x^\backslash nu)\backslash overline\; y^j$.

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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 $\backslash pi:Y\backslash to\; X$ be an affine bundle modelled on a vector bundle $\backslash overline\backslash pi:\backslash overline\; Y\backslash to\; X$. Every global section $s$ of an affine bundle $Y\backslash to\; X$ yields the bundle morphisms

$Y\backslash ni\; y\backslash to\; y-s(\backslash pi(y))\backslash in\; \backslash overline\; Y,\; \backslash qquad$

\overline Y\ni \overline y\to s(\pi(y))+\overline y\in Y.

In particular, every vector bundle $Y$ has a natural structure of an affine bundle due to these morphisms where $s=0$ is the canonical zero-valued section of $Y$. For instance, the tangent bundle $TX$ of a manifold $X$ naturally is an affine bundle.

An affine bundle $Y\backslash to\; X$ is a fiber bundle with a affine group fiber bundle $GA(m,\backslash mathbb\; R)$ of affine transformations of its typical fiber $V$ of dimension $m$. This structure group always is reducible to a general linear group $GL(m,\; \backslash mathbb\; R)$, i.e., an affine bundle admits an atlas with linear transition functions.

By a morphism of affine bundles is meant a bundle morphism $\backslash Phi:Y\backslash to\; Y\text{'}$ whose restriction to each fiber of $Y$ is an affine map. Every affine bundle morphism $\backslash Phi:Y\backslash to\; Y\text{'}$ of an affine bundle $Y$ modelled on a vector bundle $\backslash overline\; Y$ to an affine bundle $Y\text{'}$ modelled on a vector bundle $\backslash overline\; Y\text{'}$ yields a unique linear bundle morphism

$\backslash overline\; \backslash Phi:\; \backslash overline\; Y\backslash to\; \backslash overline\; Y\text{'},\; \backslash qquad\; \backslash overline\; y\text{'}^i=$

\frac{\partial\Phi^i}{\partial y^j}\overline y^j,

called the linear derivative of $\backslash Phi$.

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See also

• Fiber bundle
• Fibered manifold
• Vector bundle
• Affine space

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Notes

• 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, ; .

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