  In this paper we compute the corrections to the black-body force (BBF)
potential due to spacetime geometry and topology. This recently discovered
attractive force on neutral atoms is caused by the thermal radiation emitted
from black bodies and here we investigate it in relativistic gravitational
systems with spherical and cylindrical symmetries. For some astrophysical
objects we find that the corrected black-body potential is greater than the
flat case, showing that this kind of correction can be quite relevant when
curved spaces are considered. Then we consider four cases: The Schwarzschild
spacetime, the global monopole, the non-relativistic infinity cylinder and the
static cosmic string. For the spherically symmetric case of a massive body, we
find that two corrections appear: One due to the gravitational modification of
the temperature and the other due to the modification of the solid angle
subtended by the atom. We apply the found results to a typical neutron star and
to the Sun. For the global monopole, the modification in the black-body
potential is of topological nature and it is due to the central solid angle
deficit that occurs in the spacetime generated by that object. In the
cylindrical case, which is locally flat, no gravitational correction to the
temperature exists, as in the global monopole case. However, we find the
curious fact that the BBF depends on the topology of the spacetime through the
modification of the azimuthal angle and therefore of the solid angle. For the
static cosmic string we find that the force is null for the zero thickness
case.
