Transition functions from local sections

For each overlap $U_{ij}:=U_i\cap U_j$ and each $x\in U_{ij}$, the points $s_i(x)$ and $s_j(x)$ lie in the same fiber $P_x$. Since the right action is free and transitive on each fiber, there exists a unique element $g_{ij}(x)\in G$ such that

This defines a smooth map

called the transition function for the pair $(i,j)$. These are the principal bundle transition functions associated to the chosen local sections, and they encode the change of the local trivializations coming from the sections.

From the defining equation and uniqueness, one immediately gets:

• $g_{ii}(x)=e$ on $U_i$,
• $g_{ji}(x)=g_{ij}(x)^{-1}$ on $U_{ij}$,
• On triple overlaps $U_{ijk}=U_i\cap U_j\cap U_k$,
  $$g_{ij}(x)\,g_{jk}(x)=g_{ik}(x),$$
  which is the cocycle condition.

Changing the local sections changes the functions $g_{ij}$ by an equivalence of cocycles, leaving the underlying bundle unchanged.

1. Trivial bundle. For $P=M\times G$ with global section $s(x)=(x,e)$, any cover and the restricted sections $s_i=s|_{U_i}$ give $g_{ij}\equiv e$ on all overlaps.

1. Hopf fibration. In the Hopf bundle $S^3\to S^2$ with structure group $U(1)$, take the standard cover of $S^2\cong \mathbb{CP}^1$ by the charts $U_0=\{[z_1:z_2]\mid z_2\neq 0\}$ and $U_1=\{[z_1:z_2]\mid z_1\neq 0\}$. Using the local sections
   $$s_0(w)=\frac{(w,1)}{\sqrt{1+|w|^2}},\qquad s_1(w')=\frac{(1,w')}{\sqrt{1+|w'|^2}},$$
   with $w=z_1/z_2$ and $w'=z_2/z_1$, the overlap has $w'=1/w$ and one finds a transition function $g_{01}:U_0\cap U_1\to U(1)$ satisfying $s_1(1/w)=s_0(w)\cdot g_{01}(w)$; a concrete choice is
   $$g_{01}(w)=\frac{|w|}{w}\quad (w\neq 0),$$
   with $g_{10}=g_{01}^{-1}$.

1. Möbius bundle as a principal $O(1)$-bundle. View the Möbius line bundle over $S^1$ as a principal $O(1)=\{\pm 1\}$-bundle. With two local sections over two arcs whose overlap has two components, one component can have transition value $+1$ and the other $-1$, producing a nontrivial cocycle and hence a nontrivial bundle.