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[derralf]

I would like to simulate the chaotic flow along the geodesics of a compact Riemann Surface of constant positive negative curvature (cf. Hadamard Billard). However, I could not find out how to represent the manifold numerically. Any ideas?

Comments

[jestingrabbit]

You mean constant negative curvature.

Anyway, what you need is a model for the hyperbolic plane, which also has constant negative curvature. You then cut a regular octagon where all the angles are pi/4, and then identify the edges in such a way as to make the surface have genus 2 (so, sides 1 and 3, 2 and 4, 5 and 7 and 6 and 8 are identified).

The model of the hyperbolic plane that I usually pick is the poincare disc. Its conformal, so angles are easy to analyse. The points in the plane are the unit disc in the complex plane, the boundary is the unit circle, the geodesics are diameters of the circle and circles that are perpendicular to the boundary of the disc, and isometries are a particular subgroup of the moebius transformations (the subgroup is those transforms that map the disc to the disc, and are isomorphic to PSL(2,R)).

Anyway, remember that the geodesic flow is a map on the unit tangent bundle. Good luck.

[derralf]

thank you. I'll try to digest your advice. Since I have no background in topology, I don't understand the second part of your answer yet. But the octagon is my compact Riemann surface, right? And the geodesics don't intersect on it?

[jestingrabbit]

The octagon is the surface. In much the same way, if you take a square and identify the top and bottom sides and left and right sides, you get a torus, here you want to sew up the octagon to get a donut with two holes.

I'm not sure what you mean when you say the geodesics don't intersect on it, but it seems you're at the start of your journey on this one.

[derralf]

many thanks. I'll try the games on my windows pc later today.

[derralf]

hmm. I was hoping for a compact, borderless 2d manifold with a continuous flow field that is everywhere divergent. The surface of the two-hole donut would be perfect. But I might be completely off...
Edit : I saw it as soon as I wrote it - compact and everywhere divergent doesn't work.