论文标题
瑟斯顿几何的古典概念和问题
Classical Notions and Problems in Thurston Geometries
论文作者
论文摘要
在瑟斯顿的几何形状中，已经进行了广泛的研究，那些具有持续曲率几何形状（Euclidean $ \ euc $，Euc $ \ euc $，夸张$ \ sph $，Spherical $ \ sph $）的研究已进行了广泛的研究，但其他五个几何形状，$ \ hxr $，$ \ sxr $，$ \ sxr $，$ \ nil $，$ \ nil $，$ \ nil $，$ \ nil $ \ slr $ \ slr $ \ slr $，几何和拓扑观点。然而，可以制定经典概念，这些概念突出了这些概念的美感和潜在结构，例如地球曲线和球体，晶格，地球三角形及其表面，它们的内部角度和与恒定曲率几何形状中已知的陈述相似的陈述。这些不是关注的重点。在这项调查中，我们总结了有关此主题的结果，并提出了其他开放问题。
Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.
