球面 Delaunay 三角剖分的形变空间
The Deformation Space of Delaunay Triangulations of the Sphere
AI总结：
本文确定了单位二维球面上内接凸多面体空间和严格 Delaunay 测地三角剖分空间的拓扑，证实其与光滑微分同胚群具有相同同伦型，并构造平坦环面和凸多边形的反例说明相应空间不连通。
摘要：
本文确定了内接于单位二维球面的凸多面体空间以及单位二维球面的严格 Delaunay 测地三角剖分空间的拓扑。这些空间可视为单位二维球面的微分同胚群的离散化群。因此，很自然地猜想这些空间与其光滑对应物具有相同的同伦型。本文的主要结果证实了单位二维球面的这一猜想。该结果源于对 I. Rivin 发展的三角剖分曲面变分原理的一个观察。相反，类似的猜想在平坦环面和凸多边形的情形下并不成立。我们将构造平坦环面和凸多边形的简单例子，使得相应的 Delaunay 测地三角剖分空间不是连通的。
英文摘要：
In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin. On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.
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