Conformally Flat Hypersurfaces In Four-dimensional Lorentzian Space

In this paper,we study conformally flat hypersurfaces in four-dimensional Lorentzian space forms.Since Q14 the same couformal compactification of Lorentzi-an space forms of R14,H14,the conformal geometry of these three Lorentzian space forms are equivalent.So in the following we choose R14,as the ambient space.Using the light,cone model,we can lift,the spacelike or Lorentzian hypersurfa.ces x to the Mobius position vector of R26,then the study of conformal geometry of x under the conformal group is equivalent to that of the geometry of Y under the orthogonal group O(4,2).We note that for spacelike hypersurface in R14 the shape operator is diagonal,and for Lorentzian hypersurface the shape operator is self-adjoint to its Lorentzian metric,which has four standard forms.The paper is organized as follows:In the preface partthe research background of conformally flat hypersurfaces of four-dimensional Lorentzian space and the domestic and abroad research present situation are introduced firstly.Then the main work and organization structure are summarized.In Chapter 1,we give basic theory of hypersurfaces in R14 as the preparation for the latter study.In Chapter 2,we study the spacelike conformally flat,hypersurfaces with three distilct principal curvatures in 01 R14.We study the expressions of the Gauss equations,Codazzi equations,curvature tensor,Ricci tensor,scalar curvature,Weyl tensor and Schouten tensor of the spacelike hypersurfaces in R14 at the given frame.The structure equations of conformal invariant canonical lift are given by constructing the conformal fundamental closed form and conformal invariant curvature,and then we can calculate the integrability conditions.Thus we give 1-1 corresponding between spacelike conformally flat hypersurfaces with three distinct principal curvatures i1 R14 and solutions of a PDE system.We also give some examples of spacelike conformally flat hypersurfaces with three distinct principal curvatures i1 R14,and verify these examples satisfying the integrability conditions.In Chapter 3,we study the Type I(i.e.with three distinct principal curvatures)Lorentzian conformally flat hypersurfaces in R14.We study the expressions of the structure equations,Gauss equations,Codazzi equations,curvature tensor,Ricci tensor,scalar curvature,Weyl tensor and Schouten tensor of the Type ? Lorentzian hypersurfaces in R14 at the given frame.The structure equations of conformal in-variant,canonical lift are given by constructing the conformal fundamental closed form and conformal invariant curvature,and then we can calculate the integrability conditions.Thus we give 1-1 corresponding between the Type ? Lorentzian confor-mally flat hypersurfaces in R14 and solutions of a PDE system.We also give some examples of the Type ? Lorentzian conformally flat hypersurfa.ces in R14,and verify these examples satisfying the integrability conditions.In Chapter 4,we study the Type ?(i.e.with a pair of conjugate complex prin-cipal curvatures)Lorentzian conformally flat hypersurfaces in R14.We study the expressions of the structure equations,Gauss equations,Codazzi equations,curva-ture tensor,Ricci tensor,scalar curvature,Weyl tensor and Schouten tensor of the Type ? Lorentzian hypersurfaces in R14 at the given frame.The structure equations of conformal invariant canonical lift are given by constructing the conformal funda-mental closed form and conformal invariant curvature,and then we can calculate the integrability conditions.Thus we give 1-1 corresponding between the Type ?Lorentzian conformally flat hypersurfaces in R14 and solutions of a PDE system.We also give some examples of the Type ? Lorentzian conformally fla hypersurfaces in R14,and verify these examples satisfying the integrability conditions.In Chapter 5,we study the Type ?(i.e.whose shape operator is non-diagonaliz-able and has two distinct real eigenvalues)Lorentzian conformally flat hypersurfaces in R14 We study the expressions of the structure equations,Gauss equations,Co-dazzi equations,curvature tensor,Ricci tensor,scalar curvature,Weyl tensor and Schourten tensor of the Ttype ? Lorentzian hypersurfaces in R14 at the given frame.Using the projective light-cone model of the conformal geometry of R14,we get the theorem of complete classification of the Type ? Lorentzian conformally flat hy-persurfaces after some examples are given.In Chapter 6,we summary the whole work of this paper and make a prospect of the following studies.