Conformally Homogeneous Hypersurfaces In Four-dimensional Lorentzian Space

In this thesis,we study conformal homogeneous hypersurfaces in four dimension-al projective space Q14 and the hypersurfaces studied here are all connected regular hypersurfaces.Since Q14 is the conformal compactification of Lorentzian space forms of R14,S14,H14,the study of conformal homogeneous hypersurfaces in Q14 is equivalent to the study of conformal homogeneous hypersurfaces in R14,S14,H14.So we choose R14 as the ambient space in this thesis.By the projective light cone model,we lift spacelike and timelike hypersurfaces of R14 into the light cone C15 in R26 and the study of hypersurfaces under conformal transformation group acting is equivalent to the conformal geometry of image in the light cone under O(4,2).The thesis is organized as follows:In the preface,we introduce the background of the study of conformal homoge-neous hypersurfaces in R14 and related works on this topic,and the research contents are summarized.In chapter 1,we first introduce the projective space Q14.Then,by the method of moving frame we provide the fundamental theory for timelike hypersurfaces in R14.In chapter 2,we study the connected regular spacelike conformal homogeneous hypersurfaces in R14.First,we give the expressions of the structure equations,the Laplace operator and the standard scalar curvature for spacelike hypersurface in R14.Then,by introducing the conformal invariant metric gc,conformal invariant curvature W,the canonical lift Y,conformal tangent frames {Ei} and conformal normal frame ?,we derive a complete conformal invariant system {W;E1,E2,E3}for spacelike hypersurface.Finally,we obtain classification theorems for spacelike conformal homogeneous hypersurfaces,by providing all examples,together with the corresponding conformal transformation subgroups.In chapter 3,we study the connected regular timelike conformal homogeneous hypersurfaces in R14 with diagonalizable shape operator.First,we give the expres-sions of the structure equation,the Laplace operator and the standard scalar cur-vature for timelike hypersurface with type ? and type ?.Then,by introducing the conformal invariant metric gc,conformal invariant curvature W,the canonical lift Y,conformal tangent frames {Ei} and conformal normal frame ?,we derive a complete conformal invariant system {W;E1,E2,E3} for timelike hypersurface with type I and type II.Finally,we obtain classification theorems for timelike conformal homo-geneous hypersurfaces with type ? and type ?,by providing all examples,together with the corresponding conformal transformation subgroups.In chapter 4,we study the connected regular timelike conformal homogeneous hypersurfaces in R14 with non-diagonalizable shape operator.First,we give the struc-ture equations of the Laplace operator,the Laplace operator and the standard scalar curvature for timelike hypersurface with type ?.Then,by introducing the conformal invariant metric gc,conformal invariant curvature W,the canonical lift Y,conformal tangent frames {Ei} and conformal normal frame ?,we derive a complete confor-mal invariant system {E1,E2,E3} for timelike hypersurface with type ?.Finally,we obtain classification theorems for timelike conformal homogeneous hypersurfaces with type ? by providing all examples,together with the corresponding conformal transformation subgroups.In chapter 5,we summarize the above work in this thesis and provide a prospect for the studies in future.