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

TL;DR

This paper explores the geometry of force free Moebius motions of the circle, revealing that the space decomposes into a product involving a disc with a metric related to the 'better than nice' metric, and characterizes geodesics as hypocycloids.

Contribution

It provides a detailed analysis of the geodesics of the Moebius transformation group with a specific Riemannian metric, connecting it to the 'better than nice' metric on the disc.

Findings

Decomposition of the Moebius transformation space as a Riemannian product

Identification of the disc's metric as 'better than nice'

Geodesics are hypocycloids, described by a differential equation.

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.