  This work investigates the magnetically driven ferroelectricity in
orthorhombic manganites RMnO3 (R=Gd, Dy or Tb) from the point of view of the
symmetry. The method adopted generalizes the one used to characterize the polar
properties of displacive modulated structures to the case of an irreducible
magnetic order parameter. The symmetry conditions for magnetically induced
ferroelectricity are established and the Landau-Devonshire free energy
functionals derived from general symmetry considerations. The ferroelectric
polarisation observed in DyMnO3 and TbMnO3 at zero magnetic field is explained
in terms of the symmetry of a reducible magnetic order parameter. The
polarisation rotation induced in these compounds by external magnetic fields
and the stabilization of a ferroelectric phase in GdMnO3 are accounted for by a
mechanism in which magnetization and polarization are secondary order
parameters that are not directly coupled but compete with each other through
their coupling to competing primary modulated order parameters.
