Circle gluing, curvature and connection variations

Form the set-sized disjoint union ${\coprod }_i(U_i\times U(1))$ and identify $(x,z_i)$ with $(x,g_{ij}^{-1}(x)z_i)$ in chart $j$. Reflexivity follows from $g_{ii}=1$; the cocycle gives $g_{ji}=g_{ij}^{-1}$ and shows that transfer from $i$ to $j$ to $k$ equals direct transfer from $i$ to $k$. Hence these identifications define an equivalence relation and a well-defined projection $\pi :P\to M$.

For fixed $i$, the quotient map restricts to a bijection from $U_i\times U(1)$ onto ${\pi }^{-1}(U_i)$: every representative over $x\in U_i$ transfers uniquely to chart $i$, and a transition loop acts as the identity by the cocycle. On overlaps the chart changes are $(x,z_i)\mapsto (x,g_{ij}^{-1}(x)z_i)$, smooth diffeomorphisms with smooth inverse. The quotient map is open because the saturation of any open set is a union of its images under these overlap diffeomorphisms; therefore the bijections are homeomorphisms and form a smooth atlas.

Distinct base points have disjoint preimage neighborhoods because $M$ is Hausdorff. For two distinct points over the same base point, choose a chart containing that base point and disjoint fibre neighborhoods in $U(1)$; their products with a common base neighborhood separate the total-space points. Thus $P$ is Hausdorff. Fix one countable base $(B_n)$ of $M$ and the explicit countable base $(C_m)$ of circle arcs obtained by intersecting $U(1)\subset {\mathbb{R}}^2$ with rational open rectangles. In chart $i$, the products $(B_n\cap U_i)\times C_m$ form a base. Their images under the open charts form a base of $P$ indexed by the countable set $I\times \mathbb{N}\times \mathbb{N}$, which has an explicit finite-product enumeration. Thus $P$ is second countable, without selecting separate bases for a countable family of spaces.

Define the right action by $[x,z_i]\cdot \zeta =[x,z_i\zeta ]$. It is well-defined because the chart transition is left multiplication by $g_{ij}^{-1}$ and hence commutes with right multiplication. In each chart it is smooth; it is free and transitive on every fibre. The charts in step 2.1 are equivariant, so $P\to M$ satisfies the principal-bundle definition in [F2].

Let $\eta =-i{z}^{-1}dz$ be the real invariant one-form on $U(1)$, normalized by $\eta ({\mathrm{\partial }}_{\theta })=1$ for the local angle $z=e^{i\theta }$. In chart $i$ set ${\omega }_i=\eta +a_i$. If $z_j=g_{ij}^{-1}{z}_i$, then ${\eta }_j={\eta }_i-\lambda (g_{ij})$, while [F1] gives $a_j=a_i+\lambda (g_{ij})$; thus ${\omega }_j={\omega }_i$ on every overlap and the local forms define a global smooth one-form $\omega$. Its value on the vertical generator is $1$, it is right invariant, and in every chart $\ker {\omega }_i$ is a complement to the vertical line, since for each base tangent vector there is a unique circle tangent vector cancelling $a_i$. Hence $\omega$ is a principal connection.

In chart $i$, $d\omega =d{a}_i$ because $d\eta =0$. On overlaps $d{a}_j=d{a}_i+d\lambda (g_{ij})=d{a}_i$, since locally $\lambda (g_{ij})=d{f}_{ij}$ and $d^2{f}_{ij}=0$ by [F3]. The forms $d{a}_i$ therefore glue to a global two-form $f$, and the local identity $d\omega ={\pi }^{\ast }f$ proves the curvature is basic and closed. Multiplication by the fixed constant ${\kappa }_0$ gives $A_i={\kappa }_0{a}_i$ and $F={\kappa }_{0}f$ with $d{A}_i=F$.

If $(a_i^{\mathrm{\prime }})$ is another compatible family for the same transition maps, then $(a_j^{\mathrm{\prime }}-a_j)-(a_i^{\mathrm{\prime }}-a_i)=0$ on each overlap. Thus the differences patch to a global one-form $\alpha$ on $M$. Conversely, adding any global one-form $\alpha$ to every $a_i$ preserves $a_j=a_i+\lambda (g_{ij})$ and changes curvature from $f$ to $f+d\alpha$ without changing the transition data. This proves the fixed-bundle variation claim.