Umbilical Lorentzian Isoparametric Hypersurfaces Of Type Ⅱ In S₁ⁿ⁺¹

In this thesis,isoparametric hypersurfaces of typeâ…¡in the Lorentzian spheres S₁ⁿ⁺¹ are studied.Analytic expressions and local rigidity theorems for Lorentzian isoparametric hypersurfaces of typeâ…¡in S₁ⁿ⁺¹ are given.The paper is divided into 3 sections.In section 1,the historic background of the involved problem is presented and the main results are introduced.In section 2, isoparametric hypersurfaces in the Lorentzian sphere S₁⁵ are srudied.It is proved that any Lorentzian isoparametric hypersurfaces M of typeâ…¡in S₁⁵ is locally congruent to a parallel hypersurface of a Lorentzian isoparametric(?) with minimal polynomialλ².And(?) is determined uniquely by three functions C₁(t),C₂(t) and C₃(t).For Lorentzian isoparametric hypersurfaces(?) with minimal polynomialλ² in S₁⁵ the analytic expression is given.In section 3,the results will be extended to any dimensional umbilical Lorentzian isoparametriic hypersurfaces M of typeâ…¡in Lorentzian sphere S₁ⁿ⁺¹.It is proved that M is uniquely determined by a set of functions C₁(t),...,C_n-1(t) which can be chosen arbitrarily,if the minimal polynomial of the shape operator A of M isλ².The analytic expression for such kind of hypersurfaces is given.Consequently,any umbilical Lorentzian isoparametric hypersurfaces of typeâ…¡with minimal polynomial(λ-a)² of A are the parallel hypersurface of M.