Hyperbolic Hopf Fibrations
The Hopf Fibration of S^3 is amazing and beautiful. Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by +Niles Johnson
To understand it better (and fibrations in general), I recommend this talk by Niles too.www.youtube.com/watch?v=QXDQsmL-8Us
It turns out there is an analogue of the Hopf fibration for H^3. In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case. There is a 2-parameter family of them, plus one additional fibration that does not fit the family. As with S^3, fibers in the H^3 cases are geodesics. They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration. I found the following dissertation by Haggai Nuchi's a good intro and resource to help think about all this.www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf
I'm posting three early pictures of H^3 fibrations below, labeled with the notation of Haggai's dissertation. I plan to experiment more! I'd love to see an interactive visualization rather than still renders, and to hold some kind of 3D print in hand. Fibers don't touch each other, but perhaps an approach similar to +Henry Segerman
's S^3 models can work for a print (www.youtube.com/watch?v=fUWyHbOiwbI
I don't yet know if the fibers project down to a 2D surface, say the hyperbolic plane, like they do in the S^3 case. I bet they do. I also don't yet know what kinds of surfaces of fibers (like tori in the S^3 case) are natural here. Anyone know about these questions?