| In this thesis, isoparametric hypersurfaces of typeⅢin the Lorentzianspheres S14 and S15 are studied. Analytic expressions and local rigidity theoremsfor Lorentzian isoparametric hypersurface of typeⅢin S14 and S15 are given.The paper is divided into 3 sections. In section 1, the historic background of theinvolved problem is presented and the main results are introduced. In section 2,Lorentzian isopammetric hypersurfaces of typeⅢin S14 is studied. It is provedthat any Lorentzian isoparametric hypersurface M of typeⅢin S14 is locallycongruent to a parallel hypersurface of a Lorentzian isoparametric hypersurfacewith minimal polynomialλ3. And M is determined uniquely by threefunctions C1(u), C2(u) and C3(u). For Lorentzian isopararnetric hypersurface Mwith minimal polynomialλ3 in S14 the analytic expression is given. In section 3,Lorentzian isopammetric hypersurfaces of typeⅢin S15 is studied. It is provedthat any Lorentzian isoparametric hypersurface M1 with distinct principal curvaturesin the Lorentzian sphere S15 is locally congruent to a parallel hypersurface of aLorentzian isoparametric hypersurface M1, which is determined uniquely by fivefunctions F1(u), F2(u), F3(u), F4(u) andθ'(u). For Lorentzian isoparametric hyper-surface M1 with minimal polynomial (λ-1)3(λ+1) in S15 the analytic expressionis given.
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