SPACELIKE HYPERSURFACES IN RIEMANNIAN OR LORENTZIAN SPACE FORMS SATISFYING Lkx = Ax plus b

We study connected orientable spacelike hypersurfaces x : M-n -> M-q(n+1)(c), isometrically immersed into the Riemannian or Lorentzian space forms of curvature c = -1,0,1, and index q = 0, 1, satisfying the condition L(k)x = Ax + b, where L-k is the linearized operator of the (k + 1)th mean curvature Hk+1 of the hypersurface for a fixed integer 0 <= k < n, A is a constant matrix and b is a constant vector. We show that the only hypersurfaces satisfying that condition are hypersurfaces with zero Hk+1 and constant H-k (when c not equal 0), open pieces of totally umbilic hypersurfaces and open pieces of the standard Riemannian product of two totally umbilic hypersurfaces.