Updated: Feb 3, 2021
I've ended up thinking about the space of unoriented geodesics fairly often recently and just a week or two ago, I forgot what the topology was on it, and how to define open sets and...let's just say I had to get back to basics to figure things out.
Let N be a compact surface of negative curvature with Riemannian metric g. Let G(g) be the space of unoriented geodesics in the universal cover of N with the appropriate lift of g. Why would we want to think about this space? Well, it's interesting by itself.
We can uniquely define a geodesic by its 'endpoints' on the boundary. (If we choose the asymptotic class going forward and it's asymptotic class going backward, that leaves just one geodesic.) So one way to think about geodesics is as pairs of points on the boundary at infinity. The boundary at infinity is something that we do actually like to think about.
Also, once you start saying things like "I want to think about the geodesics passing through this set," wanting the set of all of the geodesics to be a space is right around the corner. Once we want that, it would be nice to know how to actually discuss what an open set of geodesics would be, and all of a sudden, we've got a topology.
Then, we get into the situation where a point in G(g) is actually a geodesic. Situations where I can confuse the words 'line' and 'point' are not great, so settling on a way to think about the topology was necessary for me to have a reasonable conversation about this space. The two standard options answer two natural questions. The first is below.
How do we talk about geodesics that pass through an open set in the universal cover?
So I look at the universal cover, I draw a ball through it and I want a way to describe the set of geodesics passing through the red open ball as an open set. The ball is open, so the set should be, right?
But what kinds of open sets would describe something like this?
Well, there's a topology that is totally the right to answer this question: the compact-open topology. To be honest, having the words compact and open that close to each other make me uncomfortable, but this topology is great. To make sense of it, we think about geodesics as what they really are: maps into the universal cover.
The compact open topology tells us that open sets in G(g) have the following form: Where K is compact and U is open,
Although this set is totally reasonable, and is the set of geodesics which pass through U at times contained in K, that's not often the kind of set I want to think about. We can pare this down a little bit, since points are compact in the reals. For some fixed t
This is a little better, since then, it's asking, geodesics that pass through U at time t. Now, if I want all of the geodesics that ever pass through U, I can take the union of the sets above over all real numbers t. So the thing I really wanted to be true is true: if U is an open set in the universal cover, then
is an open set. So the the idea of thinking that set of geodesics passing through an open set must also be open is right, even though at first glance, I wasn't convinced. Also, this means that the set of geodesics passing through a point (or any closed set) is also closed.
Exciting takeaway: Our intuition is pretty correct in what we'd like open and closed sets to be like in this space.
This picture gives an idea of why I was reluctant to believe this was an open set at first. These geodesics look way too different to be in an open set that's a basis element.
Sometimes pictures lead us astray, though. (At least, they lead me astray sometimes). Each geodesic in this picture is actually a point in G(g).
In a future post, I'll talk about the second topology I mentioned. That one answers the question: What if I want to think about geodesics that begin and end near each other?
Note: I thought about the compact open topology again after picking up Croke and Dairbekov's Lengths and volumes in Riemannian manifolds (Duke Math. J. 125 (2004)), and realizing that was the topology they gave in the definition of G(g).