Some Problems On Geometry Of Submanifolds In Pseudo-Riemannian Space Forms

In this paper,we study some problems on geometry of submanifolds in pseudo-Riemannian space forms. We give some characteristics and classifications of proper hyper-surfaces in pseudo-Riemannian space forms. In particular, some interesting characteristics of r-minimal hypersurfaces are obtained.On the one hand,using the method of moving frames and L_r operator, we investigate hypersurfaces x : M_sⁿ→(?) (c) immersed into a pseudo-Riemannian space form (?) (c) satisfying certain conditions on L_r operator:1. Proper hypersurfaces in a pseudo-Riemannian space form (?) (c) ((?) R_v^n+p or R_v+1^n+p),whose position vector x satisfy L_rx = Rx + b,where L_r is the linearized operator of the (r + l)-th mean curvature H_r+1 of hypersurfaces for a fixed r = 0,…,n-1, R∈R^{(n+1)×(n+1)} or R^{(n+2)×(n+2)} is a constant matrix,according to c = 0 or c≠0,respectively; and b∈R_v^n+p or R_v+1^n+p is a constant vector. If M_sⁿ satisfy one of the following properties:(1) c = 0, (2) c≠0,b = (?) and H_r is a constant, (3) c≠0, b = (?) and R is self-adjoint, then M_sⁿ are r-minimal (i.e. H_r+1 = 0) or isoparametric. In particular, we locally classify such spacelike hypersurfaces which are not r-minimal.2. Proper hypersurfaces Mⁿ in a pseudo-Riemannian space form (?) (c) satisfying certain equation on the mean curvature vector (?) of Mⁿ in pseudo-Riemannian space form (?) (c). We give some new characteristics of r-minimal hypersurfaces.3. Complete spacelike hypersurface immersed into a Lorentzian space form (?) (c) satisfying equationφ=λψfor some real number A, whereφ= andψ= <(?),a>, for some fixed nonzero vector a∈R₁ⁿ⁺¹,R₁ⁿ⁺² or R₂ⁿ⁺²,according to c = 0, c = 1 or c = -1,respectively.We prove that if Mⁿ has constant mean curvature, then Mⁿ is either a totally umbilical hypersurface or a hyperbolic cylinder.In adition. using L_r operator,we discuss stability of spacelike hypersurfaces with constant 2th mean curvature in a Lorentz manifolds.On the other hand, we investigate spacelike hypersurfaces with constant rth mean curvature and two distinct principal curvatures in a Lorentz space forms (?) (c). We obtain some characteristics and rigity results of spacelike hypersurfaces in Lorentz space forms. In particular,we classify completely constant mean curvature spacelike hypersur- faces with two distinct principal curvatures in an anti-de Sitter space H₁ⁿ⁺¹(c) and solve a open problem presented by Cao and Wei.