Affine Bundles vs Affine Spaces

Question

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. Can somebody give a quick explanation? Thx alot ;)

Answer 1

We say that a set $A$ is an affine space "modeled" on a vector space $V$ if it is equipped with a free transitive action of $V$ (with respect to its abelian group structure). The action of a vector $v\in V$ on $a\in A$ is denoted by $a+v$.

An affine map between two affine spaces $A_1, A_2$, modeled on the same vector space $V,$ is a map which commutes with the $V$-actions.

For example, you can check that if $A$ is an affine space modeled on $V$, then for each $v\in V$, $a\mapsto a+v$ is an affine map $A\to A$, and all affine maps $A\to A$ are of this form.

Also, any vector space $V$ can be considered as an affine space over itself, in an obvious way, and all affine spaces modeled on $V$ are isomorphic to this example.

We say that a fiber bundle $\mathcal A\to M$ is an affine bundle modeled on a vector bundle $\mathcal V\to M$ if for each $x\in M$, $\mathcal A_x$ (the fiber of $\mathcal A$ over $x$) is an affine space modeled on the vector space $\mathcal V_x$ (there are also some obvious regularity assumptions, like continuity, smoothness, local triviality, etc, depending on the context).

If you have two affine bundles over some manifold, modeled on the same vector bundle, you can now easily define, fiber by fiber, what is an affine map between them.

All this sounds quite trivial, so the point is to find some interesting examples that justify this formalism. My time is up, but perhaps some other participants of this forum (or you yourself) can supply us with such examples.