Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 55 / 3 / Solution

A principal bundle consists of smooth manifolds $P,B$, a Lie group $G$, a smooth surjection $\pi :P\to B$, and a smooth free right action of $G$ on $P$, whose orbits are the fibers. Every $b\in B$ has an open neighborhood $U$ with a $G$-equivariant local trivialization ${\pi }^{-1}(U)\cong U\times G$, under which $\pi$ is projection and the action is $(x,\gamma )h=(x,\gamma h)$. Equivalently a local section $s:U\to P$ writes every fiber point uniquely as $p=s(x)\gamma$. On overlaps, sections differ by smooth transition functions $s_{\beta }=s_{\alpha }{u}_{\alpha \beta }$ satisfying the cocycle identity.

A principal connection is a Lie-algebra-valued one-form $\omega$ on $P$ satisfying $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$ and $\omega ({\xi }^{\#})=\xi$ for the fundamental vector field ${\xi }^{\#}(p)={\frac{d}{dt}|}_{0}p\exp (t\xi )$. Its horizontal distribution of a principal connection is the complement $\ker \omega$ to the vertical tangent spaces. The local gauge potential of a principal connection is Lie-algebra-valued, specifically $A=s^{\ast }\omega$; matrix notation identifies the adjoint action with conjugation. The displayed connection expression is understood on each coordinate trivialization, not as a choice of a global section.