  The RKKY interaction between two impurities on a curved surface is
investigated. Practical curved sample parameters are
  considered for investigating curvatures, similar to those observed
experimentally in available samples such as graphene. For this type of
two-dimensional curved systems and for two types of Schr\"{o}dinger and Dirac
carriers, the RKKY interaction and local density of states (LDOS) are
calculated. Results show that the strength of the RKKY interaction on a curved
surface depends significantly on the nature of particles embedded on this
surface, indicating that the geometry has a different effect on the dynamics of
Schr\"{o}dinger and Dirac carriers. While the curved surface increases the RKKY
interaction strength for Schr\"{o}dinger-type carriers in comparison with the
flat sample, RKKY interaction decreases in the curved surface for Dirac-type
particles. This can be understood regrading the effective magnetic field
experienced by the Dirac particles as a result of the curved surface. There is
also a correlation between LDOS and the RKKY interaction, which could be
explained by the effect of conducting electrons density on the exchange
interaction mediated by these carriers.
