Affine bundle explained

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

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 functionsy' ^{i=} A ^{\nu)y} _{j(x} ^{j} + b ^{i(x} ^{\nu).}

There are the bundle morphisms

Y x _{X\overline} Y\longrightarrow Y, (y ^{i,} \overline y ^{i)\longmapsto} y ^{i} + \overline y ^{i,} Y x _{X} Y\longrightarrow \overline Y, (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 ^{\nu)\overline} _{j(x} 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 morphismsY\ni y\to y - s(\pi(y))\in \overline Y, \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, 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, 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', \overline y' ^{i = (\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, .
  • Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013, ; .

Notes and References

1. . (page 60)