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From Notices of the AMS

Illustration of a traveler moving along a Euclidean line.
Traveling along a Euclidean line.

Euclidean Traveller in Hyperbolic Worlds

by Hee Oh

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 traveling 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 dimension $n\ge 2$, guided by Kronecker.
  • In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund [Hed36].
  • In 1991, she then travels to a closed hyperbolic manifold of higher dimension $n\ge 3$ guided by Ratner [Rat91].
  • Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal'bo [Dal00] in dimension two in 2000, by McMullen-Mohammadi-O. [MMO16] in dimension three in 2016 and by Lee-O. [LO19] in all higher dimensions in 2019.

Rotations of the Circle

As a warm up, she will first do her exercise of jumping on the circle $\mathbb S^1$, which may be considered as the one-dimensional torus $\mathbb T^1$.

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