Research On Related Problems In Semi-Riemannian Geometry

This paper is mainly research Killing vector fields on compact Riemannian manifolds with negative scalar curvature and Semi-umbilical Lorentzian Isoparametric H ypersurfaces of Type III in S₁ⁿ⁺¹,And the main conclusions are as follows: we obtain the necessary conditions for the existence of Killing vector fields on compact Riemann manifolds with negative curvature,and obtain the existric theorem and uniqueness theorem for the type III semi umbilical Lorenz isoparametric hypersurfaces.This paper's structure is as follows:In Chapter 1,a brief introduction for the research history of killing vector fields vanishing and Semi-umbilical Lorentzian Isoparametric Hypersurfaces of Type III in S₁ⁿ⁺¹.The second chapter : we investigate killing vector fields on compact Riemannian mani-folds with negative scalar curvature.By using the Bochner method,we obtain a necessary condition of the existence of non-trivial killing vector fields on these manifolds.The third chapter :we research the semi-umbilical Lorentzian isoparametric hypersurfaces of type III in the Lorentzian sphere S₁ⁿ⁺¹.The existence theorem and uniqueness theorem for this kind of hypersurface are given,and the analytic expression of such hypersurface is provided.