a classification theorem on the lorentz hypersurfaces of minkowski space

In this paper‎, ‎we classify timelike hypersurfaces in lorentz-minkowski space‎, ‎$x:m^nightarrowl^{n+1}$‎,‎satisfying the condition $l_kx=ax+b$‎, ‎where $l_k$ is‎ the $k$th extension of laplace operator (i.e‎. ‎$delta$)‎, ‎$a$ is a constant matrix and $b$‎ ‎is a constant vector‎. ‎the condition $l_kx=ax+b$ is a new version of a well-known equation $delta x=d x$ for a real number $d$‎. ‎as an extension of takahashi s theorem we show that such a hypersurface has to be $k$-minimal or an open piece‎ ‎of $s_1^n(c)$‎, ‎$s_1^m(c)imesr^{n-m}$ or $s^m(c)imesl^{n-m}$ for some $c>0$ and $1