Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms

Let $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ be the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ labeled vertices and prescribed cone-deficits $\delta_1$, $\delta_2$, $\dots$, $\delta_N$. We show that $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ has the structure of a real hyperbolic manifold of dimension $2N-3$. When $N=4$ and $5$, we show that every surface in $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ can be decomposed into at most $2\binom{2N-2}{2}$ parallelograms, and the decomposition is invariant under the antipodal map. Using the edge-lengths of these parallelograms as coordinates, we show that the moduli space of centrally symmetric polyhedral surfaces with $8$ unlabeled vertices and cone-deficits $\frac{\pi}{2}$ is isometric to the quotient of a real hyperbolic regular ideal $5$-simplex by the dihedral group $D_6$.