Let $\pi:P\rightarrow M$ be a principal $Gl(n,\mathbb{R})$ bundle.
Given $x\in M$ there is an open set $U$ containing $x$ and a local trivialization $\pi^{-1}(U)\rightarrow U\times G$. This gives a cover $\{U_\alpha\}$ of $M$ and local trivializations $\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times G$. These in turn give what are called as transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow Gl(n,\mathbb{R})$. These transition functions determine the principal bundle.
Consider the determinant map $det:Gl(n,\mathbb{R})\rightarrow \mathbb{R}^*=Gl(1,\mathbb{R})$. This is smooth. So, is the composition $h_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow Gl(n,\mathbb{R})\rightarrow \mathbb{R}^*$. So, we have an open cover $\{U_\alpha\}$ of $M$ and maps $h_{\alpha\beta}:U_{\alpha\beta}\rightarrow Gl(1,\mathbb{R})$ satisfying the cocycle condition.
These maps define $G=Gl(1,\mathbb{R})$ bundle on $M$.
We call this the determinant bundle associated to $P\rightarrow M$.
I would like to understand what does this determinant bundle say about $P\rightarrow M$. In this question I came to know that if determinant bundle is trivial then the bundle is self dual.
Are there any such properties?
Best Answer
The determinant bundle is trivial if and only if the bundle is orientable.
The determinant bundle is a principal $\mathbb{R}$-bundle, it is trivial if and only if it has a $(\mathbb{R}^+,\times)$-reduction since $\mathbb{R}^+$ is contractible, this is equivalent that $M$ is orientable, in in fact the obstruction of the triviality of the determinant bundle is the 0-stiefel-Whithney class.
An alternative description of the first Stiefel-Whitney class