  The dark matter hypothesis, which is not called into question here, explains
why typical rotation curves of spiral galaxies do not follow a Keplerian
profile. It is however not sufficient in itself to explain why the whole matter
distribution in spiral galaxies is such that the rotation curve generally
presents a flat profile in the disk region.
  To understand this property, a model considering general relativistic effects
is developed. It is stressed that the aim is not to explain the flat rotation
curve of spiral galaxies without dark matter. More specifically, the analytical
stationary solution of an axisymmetric rotating pressureless fluid for the
linearized equations of the theory of general relativity is determined. It is
demonstrated that this solution leads to some constraints on the rotation
curve, by looking to its limit behavior when neglecting general relativistic
effects. In particular, the positiveness of the density imposes the rotation
curve to be flat in the regions where the density and general relativistic
effects are small, such as in the disk region. General relativistic effects
hence remain negligible in the disk region, but their consideration proved to
be necessary to establish the aforementioned constraint. Such constraint cannot
be derived from a Newtonian approach.
  The model is finally applied on two specific cases to demonstrate its ability
to predict the rotation curve from a typical density profile along the galactic
plane. These examples suggest that for some galaxies, general relativistic
effects can be significant close to the bulge region and should be taken into
account to have a proper understanding of their rotation curve.
