AN EXTENSION OF TAKAHASHI THEOREM ON LORENTZIAN SPACE FORMS

We study spacelike hypersurfaces in (pseudo-)Riemannian space forms Rn+1 q , Sn+1 q ½ Rn+2 q and Hn+1 q ½ Rn+2 q+1 for q = 0; 1, whose position vector field x satisfies the condition Lkx = Ax + b, where Lk 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. For every k, we show that, up to rigid motions, the only hypersurfaces in Rn+1 q (q = 0; 1) satisfying the above mentioned condition are k¡minimal hypersurfaces, open pieces of Sn(r) ½ Rn+1, Sm(r) £ Rn¡m ½ Rn+1, Hn(¡r) ½ Rn+1 1 and Hm(r) £ Rn¡m ½ Rn+1 1 . In the non-flat standard space forms Mn+1 q (c) for c = ¡1; 1 and q = 0; 1, for every k, when b = 0, the only hypersurfaces satisfying the condition are hypersurfaces with zero Hk+1 and constant Hk, open pieces of totally umbilic hypersurfaces and open pieces of the (pseudo-)Riemannian products of two certain totally umbilic hypersurfaces. When Hk is constant and b is an arbitrary vector, we show that the hypersurfaces satisfying that condition is totally umbilic hypersurfaces. This extended abstract is based on [9].