Hypersurfaces in pseudo-Euclidean spaces satisfying a linear condition on the linearized operator of a higher order mean curvature

摘要

We study hypersurfaces Msn immersed in pseudo-Euclidean spaces Rtn+1 whose position vector ψ satisfies the condition L_kψ = Aψ + b, where L_k is the linearized operator of the (k + 1)-th mean curvature of the hypersurface for a fixed k = 0, n - 1, A∈R~((n+1)×(n+1))is a constant matrix and b∈R_t ~(n+1) is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)-th mean curvature, open pieces of totally umbilical hypersurfaces S_t ~n(r) or H_(t-1) ~n(-r) (r > 0), and open pieces of generalized cylinders Run-m×S_(t-u) ~m(r) or Run-m×H_(t-u-1) ~m(-r) (r > 0), with k + 1 ≤ m ≤ n - 1.