  A galaxy is modeled as a stationary axially symmetric pressure-free fluid in
general relativity. For the weak gravitational fields under consideration, the
field equations and the equations of motion ultimately lead to one linear and
one nonlinear equation relating the angular velocity to the fluid density. It
is shown that the rotation curves for the Milky Way, NGC 3031, NGC 3198 and NGC
7331 are consistent with the mass density distributions of the visible matter
concentrated in flattened disks. Thus the need for a massive halo of exotic
dark matter is removed. For these galaxies we determine the mass density for
the luminous threshold as 10^{-21.75} kg.m$^{-3}.
