论文标题
双曲线世界中的欧几里得旅行者
Euclidean traveller in hyperbolic worlds
论文作者
论文摘要
我们将讨论各种几何空间中欧几里得线的所有可能关闭。想象一下欧几里得旅行者，他只沿着欧几里得线行进。她将前往许多不同的几何世界，我们的问题将是“她在每个世界中看到什么地方？”。这是我们欧几里得旅行者的行程：1884年，她在克罗内克（Kronecker）的指导下走向任何维度的圆环。 1936年，她前往世界，被赫德隆德（Hedlund）引导，被称为封闭的双曲线表面。然后，她在Ratner指导下的高维$ n \ ge 3 $的封闭双曲线歧管上旅行。最后，她冒险进入由达尔博（Dal'bo）以尺寸为单位的无限量的双曲线歧管，2000年$ 2 $，由麦克马伦·莫哈玛·阿玛迪（McMullen-Mohammadi-Oh）在2016年$ 3 $，在2019年在所有更高维度上。
We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be travelling to many different geometric worlds, and our question will be "what places does she get to see in each world?". Here is the itinerary of our Euclidean traveller: In 1884, she travels to the torus of any dimension, guided by Kronecker. In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund. In 1991, she then travels to a closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner. Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal'bo in dimension $2$ in 2000, by McMullen-Mohammadi-Oh in dimension $3$ in 2016 and by Lee-Oh in all higher dimensions in 2019.
