Abstract. We study hypersurfaces either in the sphere S n+1 or in the hyperbolic space H n+1 whose position vector x satisfies the condition L k x = Ax + 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+2)×(n+2) is a constant matrix and b ∈ R n+2 is a constant vector. For every k, we prove that when A is self-adjoint and b = 0, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)-th mean curvature and constant k-th mean curvature, and open pieces of standard Riemannian products of the form S m ( √ 1 − r 2 )× S n−m (r) ⊂ S n+1 , with 0 < r < 1, andwe also obtain a classification result for the case where b = 0.

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