Affine bundle

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In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.[1]

Formal definition[edit]

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

(i) All the fiber Y_x of Y are affine spaces modelled over the corresponding fibers \overline Y_x of a vector bundle \overline Y.

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

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

y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu).

There are the bundle morphisms

Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, \overline y^i)\longmapsto  y^i +\overline y^i,
Y\times_X Y\longrightarrow \overline Y,\qquad (y^i, y'^i)\longmapsto  y^i - y'^i,

where (\overline y^i) are linear bundle coordinates on a vector bundle \overline Y, possessing linear transition functions \overline y'^i= A^i_j(x^\nu)\overline y^j .

Properties[edit]

An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let \pi:Y\to X be an affine bundle modelled on a vector bundle \overline\pi:\overline Y\to X. Every global section s of an affine bundle Y\to X yields the bundle morphisms

 Y\ni y\to y-s(\pi(y))\in \overline Y, \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\to X is a fiber bundle with a general affine structure group  GA(m,\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, \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 \Phi:Y\to Y' whose restriction to each fiber of Y is an affine map. Every affine bundle morphism \Phi:Y\to Y' of an affine bundle Y modelled on a vector bundle \overline Y to an affine bundle Y' modelled on a vector bundle \overline Y' yields a unique linear bundle morphism

 \overline \Phi: \overline Y\to \overline Y', \qquad \overline y'^i=
\frac{\partial\Phi^i}{\partial y^j}\overline y^j,

called the linear derivative of \Phi.

See also[edit]

Notes[edit]

  1. ^ Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag . (page 60)

References[edit]