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

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.

Choose local sections related by $s^{\prime }=su$, where $u:U\to G$. Keeping the same point of the principal bundle requires $s\gamma =s^{\prime }{\gamma }^{\prime }$, so ${\gamma }^{\prime }=u^{-1}\gamma$. For reconstruction of a principal connection from local gauge potentials, the local principal connection form transforms asUsing $d(u^{-1})=-u^{-1}(du)u^{-1}$, computeandThe inhomogeneous terms cancel. Thus $\boxed{{\omega }^{\prime }=\omega }$: both local expressions define the same one-form on the total space. This is compatibility across trivializations, whereas $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$ describes the separate right-action equivariance of a principal connection. In particular a change of trivialization is not an assertion that the local gauge potential of a principal connection $A$ itself is unchanged.

Work over a coordinate neighborhood $U\subset B$ with coordinates $x^i$, and choose a basis $T_a$ of the Lie algebra, with $[T_a,T_b]=c^c{}_{ab}{T}_c$. Put $A=A_i^a{T}_{a}d{x}^i$. For each $T_a$ let $R_a$ be its right-invariant vector field on the fiber group. In matrix notation $R_a(\gamma )=T_a\gamma$, andThis minus sign is the right-invariant vector fields realize the opposite Lie algebra rule. It is essential here. Define the coordinate horizontal lifts of a principal connection byTheir projections are ${\pi }_{\ast }{H}_i={\partial }_i$, so they are linearly independent and there are exactly $\dim B$ of them. Also $d\gamma (H_i)=-A_i\gamma$, whenceThey therefore span the horizontal distribution of a principal connection on this trivialization.

The coefficients $A_i^a$ depend only on the base variables, so taking Lie brackets giveswhere $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i+[A_i,A_j]$ and $F=\frac{1}{2}{F}_{ij}d{x}^i\wedge d{x}^j=dA+A\wedge A$. Since the right-invariant vector fields form a basis on each fiber, $\boxed{[H_i,H_j]=0\text{ for all }i,j⟺F=0}$. This is the local coordinate version of vanishing curvature of a principal connection and, by the Frobenius theorem, integrability of the horizontal distribution.

The coordinate qualification is necessary: horizontal lifts of arbitrary noncommuting base fields need not commute even for a flat principal connection. Nor does flatness guarantee a globally defined coordinate frame or a global horizontal section; holonomy of a connection can obstruct the latter. If the requested frame were read globally, the trivial flat bundle $S^2\times G$ would already be a counterexample: restricting a global horizontal frame to the identity-fiber section would trivialize $T{S}^2$, contrary to the Hairy ball theorem. The construction on each coordinate trivialization supplies the intended result without that global claim.