The Relationship Between Isometric Sphere And Fixed Point In Quaternion Hyperbolic Space

In this thesis, based on the action of Mobius transformations and the properties of its isometric spheres, we obtain a decomposition g =tfO. By use of the relationship of the fixed points of the rotation O and g, we obtain a classification of quaternionic Mobius transformations. The isometric maps are classified as loxodromic elements,parabolic elements and elliptic elements. Due to the variety of the set of fixed point(s) of elliptic elements, elliptic elements are the most special maps among the three types. In general speaking, the properties of the orbifolds in high dimensions concern the properties of elliptic elements. We will obtain the normalize forms of elliptic elements in lower real,complex and quaternionic hyperbolic spaces and the properties of the sets of fixed point(s)of them.This thesis is arranged as follows.In Chapter one, we provide some background informations about our research and state our main results.In Chapter two, we introduce some basic material of quaternion and quaternionic matrix.In Chapter three, we introduce some basic knowledge about quaternionic Mobiustransformations, obtain a decomposition g =tfO and our main results.In Chapter four, We discuss the fixed points sets of elliptic elements on lower real hyperspace, complex hyperspace and quaternionic hyperspace respectively. At the sane time we get a classification of elliptic elements of quaternionic hyperspaces.