We check invariance of Lobachevski metric¶

Exercise I.9.5 [1] For the elliptic space ($\sigma=-1$) check that The M"obius invariant metric is \begin{equation} m(z,w)=\sinh^{-1}\frac{ | z-w|_e }{2\sqrt{\Im[z]\Im[w]}\strut}, \tag{I.9.8} \label{eq:elliptic} \end{equation} where $\Im[z]$ is the imaginary part of a complex number $z\$ and $| z|^2_e=u^2+v^2$.

Hint. One can directly or, by using CAS (see below), verify that this is a Möbius-invariant expression. Thus, we can transform (z) and $w$ to $i$ and a point on the imaginary axis by a suitable Möbius transformation without changing the metric. The shortest curve in the Riemannian metric is the vertical line, that is, $du=0$. For a segment of the vertical line, the expression \eqref{eq:elliptic} can easily be evaluated. See also [Beardon95, Thm.~7.2.1] for a detailed proof and a number of alternative expressions.

Solution. First we define the norm

from init_cycle import *
def Lnorm(p,p1,si):
    return (pow(p[0]-p1[0],2)-si*pow(p[1]-p1[1],2))/p[1]/p1[1]

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

Using vector formalism and idx

Calculate Möbius transformations of two given points

Pa=clifford_moebius_map(sl2_clifford(a,b,c,d,e),P,e)
Pb=clifford_moebius_map(sl2_clifford(a,b,c,d,e),P1,e)

Check the inavariance: the distange between the originals is equal to the distance between their images.

"Lobachevski-type metric is SL(2,R)-invariant in all EPH cases: %s " %\
(Lnorm(P,P1,sign)-Lnorm(Pa,Pb,sign)).normal().is_zero()

'Lobachevski-type metric is SL(2,R)-invariant in all EPH cases: True '

This notebook is a part of the MoebInv notebooks project [2] .

References¶

Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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