Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X39120130301Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying L_k(x)=Ax+b205223338ENF.PashaieTarbiat Modares University, IranS.M.B.KashaniTarbiat Modares UniversityJournal Article20110530We study connected <br />orientable spacelike hypersurfaces <br />$x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the <br />Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and <br />index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ <br /> where $L_k$ is the $textit{linearized operator}$ of <br />the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a <br />fixed integer $0leq k<n$, $A$ is a constant matrix and $b$ is a <br />constant vector. <br /> <br />We show that the only hypersurfaces satisfying that condition are <br />hypersurfaces with zero $H_{k+1}$ and constant $H_k$ ( when $cneq <br />0$ ), open pieces of totally umbilic hypersurfaces and open pieces <br />of the standard Riemannian product of two totally umbilic <br />hypersurfaces.http://bims.iranjournals.ir/article_338_564a4b1f52aa0e44c763a92cfc8189b5.pdf