Affine bundle

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

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) 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

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

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

References

• S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, 1996, ISBN 0-471-15733-3.
• Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; arXiv: 0908.1886.