# Vector Bundle Structure

A smooth vector bundle of rank $n$ is usually defined as a smooth map $p: E \longrightarrow B$ together with a real vector space structure on each fiber $E_b := p^{-1}(b)$ such that:

• (Local Triviality): There is an open covering $\{U_i\}_i$ of $B$ and trivialisations $\{\phi_i\}_i$ where $\phi_i: p^{-1}(U_i)\longrightarrow U_i\times \mathbb{R}^n$ is a diffeomorphism satisfying $p =pr_1 \circ \phi_i$ and $\phi_{ib}: E_b\longrightarrow \mathbb{R}^n$ is a linear isomorphism for every $b\in B$.

Alternatively, some authors (eg. S. Lang, Amiya Mukherjee, Michor P.W) define the notion of a vector bundle atlas.

A vector bundle chart is a diffeomorphism $\phi: p^{-1}(U)\longrightarrow U\times \mathbb{R}^n$ such that $p = pr_1\circ \phi$.

Two vector bundle charts $\phi_i, \phi_j$ are said to be compatible if there exist a smooth map $\phi_{ij}:U_i\cap U_j\longrightarrow GL(n, \mathbb{R})$ such that $\phi_j\circ\phi_i^{-1}(x, v) = (x, \phi_{ij}(x)\cdot v)$

Finally, a vector bundle atlas is a set $\{(U_i, \phi_i)\}_i$ of compatible vector bundle charts where $\{U_i\}_i$ is an open cover of $B$. Two vector bundle atlases are equivalent if their union is again a vector bundle atlas.

Now, one can define a smooth vector bundle of rank $n$ as a smooth map $p: E \longrightarrow B$ together with an equivalent class of vector bundle atlases. The two definitions are equivalent since:

• The local triviality axiom provide an (equivalent class of) atlas.
• An atlas provide a real vector space structure on each fiber: the only ones such that $\phi_{ib}: E_b\longrightarrow \mathbb{R}^n$ is an linear isomophism for every $i$ and $b\in B$.

A natural question is: can we have two non-equivalent atlases giving the same real vector space structure on the fibers? The answer is no, and that why the two definitions are equivalent.

Now consider a smooth map $p: E \longrightarrow B$ together with a Lie group $G$ acting on a smooth manifold $F$.

We can define the notion of a $G$-bundle atlas (with typical fiber $F$) as before by replacing $\mathbb{R}^n$ with $F$ and $GL(n, \mathbb{R})$ with $G$. An equivalent class of $G$-bundle atlas is a $G$-bundle structure and given a $G$-bundle structure we have a $G$-bundle. Thus a vector bundle is just a $GL(n, \mathbb{R})$-bundle.

Here are my questions:

• Can we, instead of giving a $G$-bundle atlas, give a structure on each fiber as in the case $G = GL(n, \mathbb{R})$?
• Is there a simple example of a data $p: E \longrightarrow B$ together with a Lie group $G$ acting on a smooth manifold $F$ with two non-equivalent $G$-bundle atlas on it? If so, could you explain the differences between the two examples.
• Could you explain why it is important to choose a $G$-bundle atlas; what is the meaning of this structure?

In the case of $G = GL(n, \mathbb{R})$ I am satisfied because it's equivalent with the "vector space structure on each fiber" definition but I am confused with the general case.

Thank you for helping!

Paul.

• Are you talking about the concept of "Principal bundle"? Could you please more explain on your question? – Ali Taghavi Jan 6 '17 at 14:32
• In this setting a principal bundle is when $F = G$ and $G$ act on the left on itself. – paeolo Jan 6 '17 at 14:48
• To be more precise, i would like an example as explained in the second point – paeolo Jan 6 '17 at 15:17

Consider the torus $T^2$, its tangent bundle $p:TT^2\rightarrow T^2$ is an $R^2$-differentiable bundle.You can also see this bundle as an $O(n)$-bundle by defining differentiable metrics on $T^2$, here $G=O(2)$ acts on $R^2$. There exist flat and non flat differentiable metrics on the torus which define non equivalent $O(2)$-reductions of the tangent bundle.