2019/02/13 by Rassias, Themistocles M., Suksumran, Teerapong
#15A66 #20N05 #51F15 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 51B10 #Secondary 46T99
paper · doi:10.48550/arxiv.1902.05003
The open unit ball \mathbbB = \v∈ℝn\colon‖v‖<1\ is endowed with Möbius addition ⊕M defined by u⊕Mv = \dfrac(1 + 2\langleu,v⟩ + ‖v‖2)u + (1 - ‖u2)v1 + \langleu,v⟩ + ‖u‖2‖v‖2‖ for all u,v∈ B. In this article, we prove the inequality \dfrac‖u‖-‖v‖1+‖u‖‖v‖≤ ‖u⊕M v‖ ≤ \dfrac‖u‖+‖v‖1-‖u‖‖v‖ in \mathbbB. This leads to a new metric on \mathbbB defined by dT(u,v) = tan-1‖-u⊕Mv‖, which turns out to be an invariant of Möbius transformations on ℝn carrying \mathbbB onto itself. We also compute the isometry group of (\mathbbB, dT) and give a parametrization of the isometry group by vectors and rotations.