When does the Fibre Bundle have Discrete Structure Group?

Question

Let $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ be a Fibre Bundle ((where $\mathbb{E}$ is the Total Space, $\mathbb{B}$ the Base, $\mathbb{F}$ the Fibre and $\pi: \mathbb{E} \longrightarrow \mathbb{B}$ the Projection that is a surjective submersion)).

When do we say that the fibre bundle $(\mathbb{E}, \pi, \mathbb{B}, \mathbb{F})$ has discrete structure group?

Answer 1

The fibre bundle has a discrete structure group if it is defined by a trivialization $(U_i,f_{ij}$ such that $f_{ij}:U_i\cap U_j\times F\rightarrow U_i\cap U_j\times F$ is defined by $g_{ij}(x,y)=(x,h_{ij}(y))$ where $h_{ij}$ is an automorphism of $F$. Note that in general $h_{ij}$ depends on $x$, for a discrete fibre bundle, $h_{ij}$ is defined by a constant map $U_i\cap U_j\rightarrow Diff(F)$.