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

Abstract

We study hypersurfaces Msn immersed in pseudo-Euclidean spaces Rtn+1 whose position vector ψ satisfies the condition Lkψ=Aψ+b, where Lk 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∈Rtn+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 Stn(r) or Ht−1n(−r) (r>0), and open pieces of generalized cylinders Run−m×St−um(r) or Run−m×Ht−u−1m(−r) (r>0), with k+1⩽m⩽n−1.