We have the following

1. (1)

   On $\pi^{1}_{\beta}(U^{1}_{\beta,t}\cap U^{1}_{\beta^{\prime},t})$, we write $\pi^{1}_{\beta^{\prime}}\circ(\pi^{1}_{\beta})^{-1}(\bm{x})=\bm{x}^{\prime}$. Suppose $\bm{x}$ and $\bm{x}^{\prime}$ have coordinates given by $(v_{2},0,u_{i})$ and $(v_{2}^{\prime\prime},0,u_{i}^{\prime\prime})$ in the chart $U^{1}_{\beta,0}\cap Y_{1}$. Then we have

   (7.100)

   $$\begin{cases}v_{2}^{\prime\prime}=v_{2}\cdot(1+v_{1}F_{2})\\ u_{i}^{\prime\prime}=u_{i}+v_{1}G_{i},\end{cases}$$

   where $F_{2}$ and $G_{i}$ are smooth functions in $v_{1},v_{2},u_{i}$, and $v_{1}$ is implicitly determined by $v_{2},u_{i}$ and $t$ by the equation (7.22)

2. (2)

   On $\pi^{\mathcal{N}}_{\beta}(U^{1}_{\beta,t}\cap U^{1}_{\beta^{\prime},t})$, we write $\pi^{\mathcal{N}}_{\beta^{\prime}}\circ(\pi^{\mathcal{N}}_{\beta})^{-1}(q)=q^{\prime}$. Suppose $q$ and $q^{\prime}$ have coordinates given by $(0,\zeta_{3},u_{i})$ and $(0,\zeta_{3}^{\prime\prime},u_{i}^{\prime\prime})$ in the chart $U^{1}_{\beta,0}\cap\mathcal{N}$. Then we have

   (7.101)

   $$\begin{cases}\zeta_{3}^{\prime\prime}=\zeta_{3}\cdot(1+v_{2}\tilde{F}_{3})\\ u_{i}^{\prime\prime}=u_{i}+v_{2}\tilde{G}_{i},\end{cases}$$

   where $\tilde{F}_{2}$ and $\tilde{G}_{i}$ are smooth functions in $v_{1},v_{2},u_{i}$, and $v_{1}$ is implicitly determined by $\zeta_{3},u_{i}$ and $t$ by the equation (7.22).