  A fully general relativistic description of the pulsar magnetosphere is
provided. To be more concrete, a study of the pulsar magnetosphere is performed
in the context of general relativistic magnetohydrodynamics (MHD) employing the
so-called Grad-Shafranov approach. Not surprisingly, the resulting
Grad-Shafranov equations and all the other related general relativistic MHD
equations turn out to take essentially the same structures as those for the
(rotating) black hole magnetosphere. Other different natures between the two
cases including the structure of singular surfaces of MHD flows in each
magnetosphere are essentially encoded in the different spacetime (metric)
contents. In this way, the pulsar and the black hole magnetospheres can be
described in an unified fashion. Particularly, the direction of poloidal
currents circulating in the neutron star magnetosphere turns out to be the same
as that of currents circulating in the black hole magnetosphere which, in turn,
leads to the pulsar and the black hole spin-downs via the ``magnetic braking''.
