Existence of a smooth principal connection on a compact SU(N) bundle

By [F1], choose an open trivializing cover of $M$. Compactness gives a finite subcover $(U_i{)}_{i=1}^s$ with corresponding equivariant charts $P{\mathrm{\mid }}_{U_i}\cong U_i\times SU(N)$.

In each product chart, write a point as $(x,g)$ and a tangent vector as $(v,W)$, with $W\in T_{g}SU(N)$. Define ${\omega }_i(v,W)=g^{-1}W$. This is smooth and $\mathfrak{s}\mathfrak{u}(N)$-valued because left translation identifies $T_{g}SU(N)$ with $T_{e}SU(N)=\mathfrak{s}\mathfrak{u}(N)$. For $X\in \mathfrak{s}\mathfrak{u}(N)$, the supplied Lie-group exponential ${\exp }_G(tX)$ is a curve in $SU(N)$ with derivative $X$ at $t=0$. Thus $g{\exp }_G(tX)$ has tangent $gX$ and ${\omega }_i(X_P)=X$. For fixed $h\in SU(N)$, right multiplication sends $W$ to $Wh$, and $(gh{)}^{-1}(Wh)=h^{-1}(g^{-1}W)h$; hence $R_h^{\ast }{\omega }_i={\mathrm{Ad}}_{h^{-1}}{\omega }_i$.

The finite cover in step 1.1 is an open cover of a smooth manifold. By [A1], a choice function exists for every family of nonempty sets, hence for every countable family; this is Countable Choice [F4]. Therefore [F3] and [F5] give a smooth subordinate partition $({\rho }_i{)}_{i=1}^s$ with $\mathrm{supp}{\rho }_i\subset U_i$ and ${\sum }_i{\rho }_i=1$.

Extend each ${\tilde{\omega }}_i=({\rho }_i\circ \pi ){\omega }_i$ by zero outside $P{\mathrm{\mid }}_{U_i}$. The support condition in [F5] makes each extension smooth. Set $\omega ={\sum }_{i=1}^s{\tilde{\omega }}_i$, a finite smooth sum. Since ${\rho }_i\circ \pi$ is invariant under right multiplication and each ${\omega }_i$ is right equivariant, $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$. For each fundamental vertical vector, $\omega (X_P)={\sum }_i{\rho }_{i}X=X$ by [F5]. Thus [F2] makes $\omega$ a smooth principal connection form on $P$.