Transition functions from local sections

Let $\pi:P\to M$ be a principal G-bundle with the standard right G-action . Let $\{U_i\}$ be an open cover of $M$ and let $s_i:U_i\to P$ be smooth local sections.

Construction

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 .

Basic identities

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.

Examples

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.

2. 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 $g_{10}=g_{01}^{-1}$.

3. 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.