$3$ -球面中的凸多面体与$2$ -球面的拼接
Convex Polyhedra in the $3$-Sphere and Tilings of the $2$-Sphere
摘要：
我们证明了对于$S^3$中的每个凸多面体球$P$，存在两个正则的，非边到边的$S^{2}$平铺，其平铺由$P$和对偶凸多面体球$P^*$到$P$的所有面给出.在$S^{3}$与李群$SU(2)$，$S^{2}$与$SU(2)$的李代数$su(2)$中的单位球面的同一性下，考虑$P$的向外单位法向量集$\widetilde P$和由$SU(2)$上的左右Maurer-Cartan形式定义的从$\widetilde P$到$S^{2}$的映射，得到了我们的结果.
英文摘要：
We show that for every convex polyhedral sphere $P$ in $S^3$, there exist two canonical, non-edge-to-edge tilings of $S^{2}$ whose tiles are given by all the faces of $P$ and the dual convex polyhedral sphere $P^*$ to $P$. Under the identifications of $S^{3}$ with the Lie group $SU(2)$, and of $S^{2}$ with the unit sphere in the Lie algebra $su(2)$ of $SU(2)$, our result is obtained by considering the set $\widetilde P$ of outward unit normal vectors to $P$ and the maps from $\widetilde P$ to $S^{2}$ defined by using the left and right Maurer-Cartan forms on $SU(2)$.
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