Geodesics in the hyperbolic plane (constant negative curvature K = −1)
Geodesics: 0
In hyperbolic geometry, geodesics appear as circular arcs perpendicular to the boundary of the Poincaré disk. The geodesic flow is ergodic and mixing — any trajectory eventually comes arbitrarily close to any point and direction, as proven by Eberhard Hopf in 1939 for surfaces of negative curvature.