A Complete Classification For Lorentzian Isoparametric Hypersurfaces Of Type â…¡ In S₁ⁿ⁺¹

Semi-Riemannian manifold, or called as pseudo-Riemannian manifold, is a differentiable manifold equipped with a metric of index v. When v=1it is called a Lorentzian manifold particularly, and the metric of a Lorentzian manifold is called a Lorentzian metric. The Lorentzian space form is a simply connected Lorentzian manifold with constant sectional curvature.Let M be a hypersurface immersed in a Lorentzian space form. The hypersurface M is Lorentzian if the induced metric is Lorentzian by definition. M is a Lorentzian isoparametric hypersurface if M is Lorentzian and the minimal polynomial of the shape operator A of M is constant, that is, the minimal polynomial of A is the same at each point p∈M.In the present paper the Lorentzian isoparametric hypersurfaces M of type Ⅱ in the Lorentzian sphere S1n+1(?) R1n+2are studied. The existence theorem and local rigidity theorem for those hypersurfaces are given. The hypersurface M is totally umbilical if all the principal curvatures of M equal to each other. Suppose M has two distinct principal curvatures α1,αn(α1≠αn) and the minimal polynomial of the shape operator A of M is (λ-α1)2(λ-αn). The hypersurface M is semi-umbilical if the multiplicity of α, is p=2. It is proved that M can be obtained by parallel translation of product S+p-2(t)×Sn-p(t) of two manifolds along each line Lt in a family{Lt|t∈I} of1-parameter light-like lines. Particularly M is totally umbilical if p=n, and M is semi-umbilical if p=2.In Section1the historical background and a survey of the problem studied in this paper are stated. The basic formulae for Lorentzian isoparametric hypersurface M in the Lorentzian sphere S1n+1are given, and for later use a Lemma is proved in Section2. The existence theorem for Lorentzian isoparametric hypersurface M in the Lorentzian sphere S1n+1is proved in Section3. The basic formulae appeared in Section2have been simplified in Section4by choosing suitable local frame and local coordinate. Then the local rigidity theorem for such as hypersurface is proved in Section5. In the last Section6some examples of Lorentzian isoparametric hypersurfaces in S1n+1are provided by solving the Cauchy problem of a system of ordinary differential equations with constant coefficient. One of these examples shows that a Lorentzian hypersurface in S1n+1with constant principal curvatures has not necessarily to be isoparametric, which is a phenomenon different from the isoparametric hypersurfaces in Riemannian space forms.