Affine bundle

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

Formal definition

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

(i) Every fiber ${\displaystyle Y_{x}}$ of ${\displaystyle Y}$ is an affine space modelled over the corresponding fibers ${\displaystyle {\overline {Y}}_{x}}$ of a vector bundle ${\displaystyle {\overline {Y}}}$.

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

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

${\displaystyle y'^{i}=A_{j}^{i}(x^{\nu })y^{j}+b^{i}(x^{\nu }).}$

There are the bundle morphisms

${\displaystyle Y\times _{X}{\overline {Y}}\longrightarrow Y,\qquad (y^{i},{\overline {y}}^{i})\longmapsto y^{i}+{\overline {y}}^{i},}$

${\displaystyle Y\times _{X}Y\longrightarrow {\overline {Y}},\qquad (y^{i},y'^{i})\longmapsto y^{i}-y'^{i},}$

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

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

${\displaystyle Y\ni y\to y-s(\pi (y))\in {\overline {Y}},\qquad {\overline {Y}}\ni {\overline {y}}\to s(\pi (y))+{\overline {y}}\in Y.}$

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

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

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

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

called the linear derivative of ${\displaystyle \Phi }$.

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, ISBN 0-471-15733-3.
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2013-05-28
• 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.
• Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7