A dynamical presentation of the better than nice metric on the disc

Abstract

This is a postprint of our paper "Force free Moebius motions of the circle" (J. Geom. Symmetry Phys. 27 (2012) 59-65), which we hadn't uploaded to arXiv previously. We would like to draw attention to the relationship with the article "A geometry where everything is better than nice", by Larry Bates and Peter Gibson (to appear in Proc. Amer. Math. Soc.). In our note we treat thoroughly a simple particular case of two previous, more substantial articles. We describe the force free Moebius motions of the circle, that is, the geodesics of the Lie group M ~ PSL_2(R) ~ O_o(1,2) of Moebius transformations of the circle, equipped with the Riemannian metric given by the kinetic energy induced by the action. It turns up that M decomposes as a Riemannian product S x D, where D is the unit disc endowed with a certain metric, which we now recognize as being essentially the one that is better than nice. Concerning geodesics, we only give the differential equation for their trajectories (using Clairaut's Theorem); we took pleasure in learning from the article by Bates and Gibson that they are actually hypocycloids.