Semi-umbilical Lorentzian Isoparametric Hypersurfaces Of Type Ⅱ In S₁ⁿ⁺¹

In this thesis, semi-umbilical Lorentzian isoparametric hypersurfaces of TypeⅡin the Lorentzian sphere Sln+1 are studied. The existence theorem and local rigidity theorem for semi-umbilical Lorentzian isoparametric hypersurfaces of typeⅡin Sln+1 are given.The paper is divided into 6 sections. In section 1, the historic background of the involved problem is presented and the main results are stated. To prove the existence theorem, in section 2 curves on the light cone are studied. Normal frames for these curves are constructed, and an important result is given for later use. In section 3 the existence theorem for semi-umbilical Lorentzian isoparametric hypersurface M of typeⅡwith minimal polynomial (λ-1)2(λ+1) in Sln+1 is proved. It is proved in section 4 that the two distinct principal curvature a0,a1 of M satisfy equation a0a1=-1. Therefore by consider the parallel hypersurface of M we may set a0=1,a1=-1. Then the basic formulae and the structure equations for such a hypersurface M are given in section 4. In section 5 the basic formulae and the structure equations given in section 4 are simplified by choosing suitable local coordinate and local frame. Lastly, the uniqueness theorem is proved for semi-umbilical Lorentzian isoparametric hypersurface M of type II with minimal polynomial(λ-1)2(λ+1) in Sln+1.