Isoparametric Hypersurfaces In Lorentzian Space Forms

In this thesis, isoparametric hypersurfaces in Lorentzian space forms are studied. Parametrization and local rigidity theorem of a class of Lorentzian isoparametric hypersurfaces of type â…¡ in S₁⁴ aregiven. Later, Lorentzian isoparametric hypersurfaces of type â…£ in S₁ⁿ⁺¹ are studied, and it is proved that none of these hypersurfaces have three distinct principal curvatures .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, Lorentzian isoparametric hypersurfaces of type â…¡ in S₁⁴ are studied. It is proved that any Lorentzian isoparametric hypersurface with minimal polynomial (λ-a)²(λ-a₁) in the de Sitter space S₁⁴ is locally congruent to a parallel hypersurface of a Lorentzian isoparametric hypersurface, which is determined uniquely by three functions A(u), B(u) and C(u). For Lorentzian isoparametric hypersurface with minimal polynomial (λ-1)²(λ + 1) in S₁⁴ the analytic expression is given. In section 3, it is proved that none of Lorentzian isoparametric hypersurfaces of type â…£ in S₁ⁿ⁺¹ have three distinct principal curvatures . Consequently there is no hypersurface of type â…£ in S₁⁴.