چكيده فارسي :
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].
