Let $(\mathcal{M},g_{j})$ be a sequence with gluing parameters $\beta_{j}\to\infty$. Let $\omega\in\Omega^{1}(\mathcal{M})$, then are $1$-forms $\theta_{j}^{x}$, $\theta_{j}^{y}$, $\theta_{j}^{z}$ and $\theta_{j}^{c}$ in each circle bundle part with

(7.108)

$\displaystyle|\theta_{j}^{x}|_{\tilde{g}_{j}}=|\theta_{j}^{y}|_{\tilde{g}_{j}}=|\theta_{j}^{z}|_{\tilde{g}_{j}}\to 1,\ |\theta_{j}^{t}|_{\tilde{g}_{j}}\to 0.$

Moreover, every $1$-form $\omega\in\Omega^{1}(\mathcal{M})$ in the circle bundle part can be represented as

(7.109)

$$\omega=f_{x}\theta_{j}^{x}+f_{y}\theta_{j}^{y}+f_{z}\theta_{j}^{z}+f_{t}\theta_{j}^{t}.$$