 $\mathcal{V}_{g^+}\cap\mathcal{V}_{g^-}$ is empty. We denote the sets $\mathcal{V}_{g^\pm} \cap \Lambda_\infty\Gamma$ respectively by $\mathcal{U}_{g^\pm}$, the open subset of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ corresponding to the open set $\mathcal{U}_{g^-} \times \mathcal{U}_{g^+} \times \mathbb{R}$ by $\mathcal{U}_{g}$ and the open set $\mathtt{N}(\mathcal{U}_{g})$ around $\mathtt{N}(g)$ by $\mathcal{U}_{\mathtt{N}(g)}$. Now we consider the chart $\left(\mathcal{U}_{\mathtt{N}(g)}, {\amalg}_{\mathtt{N}(g)}\right)$ and notice that $$p^{