Английская Википедия:Affine bundle
Шаблон:Short description In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.[1]
Formal definition
Let <math>\overline\pi:\overline Y\to X</math> be a vector bundle with a typical fiber a vector space <math>\overline F</math>. An affine bundle modelled on a vector bundle <math>\overline\pi:\overline Y\to X</math> is a fiber bundle <math>\pi:Y\to X</math> whose typical fiber <math>F</math> is an affine space modelled on <math>\overline F</math> so that the following conditions hold:
(i) Every fiber <math>Y_x</math> of <math>Y</math> is an affine space modelled over the corresponding fibers <math>\overline Y_x</math> of a vector bundle <math>\overline Y</math>.
(ii) There is an affine bundle atlas of <math>Y\to X</math> whose local trivializations morphisms and transition functions are affine isomorphisms.
Dealing with affine bundles, one uses only affine bundle coordinates <math> (x^\mu,y^i) </math> possessing affine transition functions
- <math>y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu).</math>
There are the bundle morphisms
- <math>Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, \overline y^i)\longmapsto y^i +\overline y^i,</math>
- <math>Y\times_X Y\longrightarrow \overline Y,\qquad (y^i, y'^i)\longmapsto y^i - y'^i, </math>
where <math>(\overline y^i)</math> are linear bundle coordinates on a vector bundle <math>\overline Y</math>, possessing linear transition functions <math>\overline y'^i= A^i_j(x^\nu)\overline y^j </math>.
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 <math>\pi:Y\to X</math> be an affine bundle modelled on a vector bundle <math>\overline\pi:\overline Y\to X</math>. Every global section <math>s</math> of an affine bundle <math>Y\to X</math> yields the bundle morphisms
- <math> 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. </math>
In particular, every vector bundle <math>Y</math> has a natural structure of an affine bundle due to these morphisms where <math>s=0</math> is the canonical zero-valued section of <math>Y</math>. For instance, the tangent bundle <math>TX</math> of a manifold <math>X</math> naturally is an affine bundle.
An affine bundle <math>Y\to X</math> is a fiber bundle with a general affine structure group <math> GA(m,\mathbb R) </math> of affine transformations of its typical fiber <math>V</math> of dimension <math>m</math>. This structure group always is reducible to a general linear group <math>GL(m, \mathbb R) </math>, i.e., an affine bundle admits an atlas with linear transition functions.
By a morphism of affine bundles is meant a bundle morphism <math>\Phi:Y\to Y'</math> whose restriction to each fiber of <math>Y</math> is an affine map. Every affine bundle morphism <math>\Phi:Y\to Y'</math> of an affine bundle <math>Y</math> modelled on a vector bundle <math>\overline Y</math> to an affine bundle <math>Y'</math> modelled on a vector bundle <math>\overline Y'</math> yields a unique linear bundle morphism
- <math> \overline \Phi: \overline Y\to \overline Y', \qquad \overline y'^i=
\frac{\partial\Phi^i}{\partial y^j}\overline y^j, </math>
called the linear derivative of <math>\Phi</math>.
See also
Notes
References
- S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, 1996, Шаблон:ISBN.
- Шаблон:Citation
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, Шаблон:ISBN; Шаблон:ArXiv.
- Шаблон:Citation
- ↑ Шаблон:Citation. (page 60)
