Some Questions On Higher Dimensional Mobius Transformations Group

M(o|¨)bius transformation in higher dimension was studied in this article andsome conclusions were drawn as follows.The background and significance of this research were introduced in thefirst chapter as well as some relevant basic knowledge on M(o|¨)biustransformation in higher dimension. In addition the main points andinnovations were also proposed.The judgment rules for strictly parabolic element and hyperbolic elementin M(o|¨)bius transformation in higher dimension was established respectively aswell as the expression for its corresponding fixed points by using Cliffordmatrix presentation in chapter2. Some characters about,-were obtainedby the judgment rule above when was hyperbolic and strictly parabolic.The Jφrgensen inequalities were proposed by Jφrgensen in1976whichwas very important for judging the discretion of M(o|¨)bius transformationgroups. The requisite condition was established in chapter3that the discretegroup generated by two elements in higher dimension was not elementary. Andthe Jφrgensen inequalities were extended into spaces of higher dimension.Chordal metric is one of the important metrics in mathematics because itcan give the distance between infinite point and limited point. Moreover, thisdistance is limited. The M(o|¨)bius transformation about chordal metricinequalities in limited dimension was extended into higher dimension inchapter4.