Affine Bundle

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

Formal definition

Let $\overline\pi:\overline Y\to X$ 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) Every fiber $Y_x$ of $Y$ is an affine space 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

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

:$\overline \Phi: \overline Y\to \overline Y', \qquad \overline y'^i= \frac\overline y^j,$

called the ''linear derivative'' of $\Phi$.

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, .
*
* Sardanashvily, G., ''Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory'', Lambert Academic Publishing, 2013, ; .
* {{citation, last1 = Saunders, first1 = D.J., title = The geometry of jet bundles, year = 1989, publisher = Cambridge University Press, isbn = 0-521-36948-7, url-access = registration, url = https://archive.org/details/geometryofjetbun0000saun

Differential geometry
Fiber bundles