It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres

$$F=\pi^{-1}(x), \ \ \ x\in B $$

over any $x\in B$, are diffeomorphic to a vector space $V$. On the other hand, a $G$-bundle over $B$, or a principal $G$-bundle is a a fibre bundle $\pi : P\to B$ over $B$, with a right free action of a Lie group $G$ on $P$ such that for any open set $U\subset B$, the locally trivial fiabrations defined by:
$$
\Phi_{U} : \pi^{-1}(U)\to U\times G, \ \ \Phi_{U}(p)=(\pi(p), \varphi_{U}(p)).
$$
Here
$$
\ \varphi_{U} :\pi^{-1}(U)\to G
$$
is a $G$-equivariant map, that is $\varphi_{U}(pg)=\varphi_{U}(p)g$, for all $p\in\pi^{-1}(U)$ and $g\in G$. In the last case the fibers are submanifolds of $P$ which are always diffeomorphic with the structure group $G$.

Although for any vector bundle $\pi : E\to B$, its fibers $F\cong V$ can be considered as Lie group with operation the vector addition, in general we **do not** include the vector bundles
as examples of principal $G$-bunldes (Although, to every vector bundle we can associate the frame bundle which is a ${\rm GL}_{n}\mathbb{R}$-principal bundle, **but I dont speak here about associated bundles**).

My question is about a good explanation about the fact that **IN GENERAL** vector bundles (themselves) **do not give examples of principal $G$-bundles**. For example, the tangent bundle $TM$ of a smooth manifold is a prototype example of a vector bundle, but itself it cannot be considered as a principal bundle for a Lie group $G$, is this true?
Thus I aks:

Which is the basic difference between a vector bundle an a $G$-bundle, which does not allows us (almost always??), to consider the vector bundles **themselves** as examples of $G$-bundles?

For example, a $G$-bundle is **trivial** (isomorphic to the product bundle), if and only if it admits a **global section**, but I think that this **is not true for vector bundles**.

A second question is about **examples of vector bundles** which can be considered the same time as $G$-bundles for some Lie group (I think that such an example is a cyllinder)

Thank you very much for your attention!

affine bundles(see e.g. the discussion in mathoverflow.net/questions/54502/affine-bundles-over-varieties). Now, an affine bundle could be principal, with the structure group $G\cong {\mathbb R}^n$. – Misha Apr 1 '12 at 20:56