Geometrical Markov coding of geodesics on surfaces of constant negative curvature

Regular minimal nets on surfaces of constant negative curvature

The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been formulated in [3], [4] in the context of the famous Plateau problem in the one-dimensional case. In [6] an asymptotic for log ♯(W (g)) as g → +∞ where g is genus and W (g) is the set of regular single-face closed local minimal nets on surfaces of curvature −1 has been obtained. It has ...

On the Measure of the Non-Special Geodesics on a Surface of Constant Negative Curvature.

A hyperbolic geometry in ' is thereby defined, the straight lines of which are segments of circles orthogonal to E. These will be called NE (noneuclidean) straight lines, and that part of a NE straight line with one endpoint in I and the other end-point on E will be called a NE ray. As is well known, there exists within I a regular non-euclidean polygon S of 4p sides, p > 1, with sides and vert...

Poleni Curves on Surfaces of Constant Curvature

In the euclidean plane, a regular curve can be defined through its intrinsic equation which relates its curvature k to the arc length s. Elastic plane curves were determined this way. If k(s) = 2α cosh(αs) , the curve is known by the name “la courbe des forçats”, introduced in 1729 by Giovanni Poleni in relation with the tractrix [9]. The above equation is yet meaningful on a surface if one int...

Hyperbolic Billiards on Surfaces of Constant Curvature

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with positive Lyapunov exponents on the sphere and on the hyperbolic plane.

Surfaces of Constant Mean Curvature