Last time, I figured out the compact open topology on G(g). As a reminder of our setting here: Let N be a 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.
The compact open topology tells us that the set of geodesics passing through an open set in universal cover is also an open set. Now I have another question we can ask about G(g) that leads to another topology.
How do we think about geodesics that begin and end near each other?
I would want it to be true that if geodesics that 'begin' and 'end' near each other then I could talk about that in terms of the topology on this space.
If these blue lines are open sets on the boundary, I'd really like the geodesics going between these sets to also be an open set.
I have those words in quotes because geodesics in the universal cover don't really begin and end anywhere. If you pick a point on a unit speed geodesic, it has infinite distance to travel both forward and backward.
Yet, stubbornly, my eyes say that a set of geodesics defined by endpoints that are near each other should be an open set.
This picture does give us a hint of the way to think about this. These blue sets are on the boundary, which isn't actually included in the universal cover. The boundary of the universal cover has an actual definition, which I'll get to after a little more discussion. We need a little more information about the behavior of geodesics first.
Definition: Two geodesics in the universal cover are (forward) asymptotic if there exists some constant C and time t such that
That is, two geodesics are (forward) asymptotic if at some point in time, they become and remain a bounded distance apart.
Being asymptotic is an equivalence class on the space of geodesics, and we then make the following definition:
Definition: The boundary at infinity is the set of asymptotic equivalence classes.
Turns out, the boundary at infinity is diffeomorphic to the unit circle. Also, given any two negatively curved metrics on our surface, the boundaries of their respective universal covers are diffeomorphic as well. They are also all isomorphic to the limit of the fundamental group of our surface N. This matches up neatly with wanting the boundary to be the circle we use to denote the universal cover.
Now, you may notice that I've slipped into talking about oriented geodesics when I said that the space I was concerned about was of unoriented geodesics. That's still what I want to do! I don't want to double count geodesics because of orientation.
Each oriented geodesic can be uniquely defined by its forward and backward asymptotic class. (If we assume there is another geodesic that has both the same forward and backward class, then we have a geodesic that is a bounded distance both going forward and backward - so they must actually be the same.) This pair of asymptotic classes can be thought of as a pair of points on the boundary. If we reverse the pair of points, then we just have the same geodesics but in the opposite direction.
That means that if we want to think about the space of unoriented geodesics, we can think about the set as pairs of points on the boundary without the diagonal (because it's not a geodesics if it begins and ends at the same point on the boundary), and quotiented out by ordering.
The space of unoriented geodesics, G(g) is diffeomorphic to
That is, G(g) can be thought of as pairs of distinct points on the boundary without worrying about the order of the pairs.
Why did I go through all of that?
Well, now we know that defining a set of geodesics by open sets on the boundary is a legitimate thing to do. The boundary of the universal cover is diffeomorphic to the unit circle, and a basis for the topology on the unit circle is open intervals of the real line (mod 1). So and open interval on the boundary defines an open set of (forward) asymptotic classes. If we take two open intervals on the the boundary, A and B, the following set is open:
In the above, think of A and B as asymptotic classes. Really the set above says that we want all the geodesics with one end in A and the other in B. (By 'end' here, I technically mean an asymptotic class.)
Exciting Takeaway: If we think about one of the blue lines on the boundary as being A and the other being B, then this picture from earlier is an open set of geodesics.
More on the boundary at infinity: Patrick Eberlein's Geometry of Nonpositively Curved Manifolds. (Around section 1.7)
More on the space of geodesics: Amie WIlkinson's Lectures on Marked Length Spectrum Rigidity. (Lecture 2, Section 2)