  Magnetism at the nanoscale has been a very active research area in the past
decades, because of its novel fundamental physics and exciting potential
applications. We have recently performed an {\it ab intio} study of the
structural, electronic and magnetic properties of all 3$d$ transition metal
(TM) freestanding atomic chains and found that Fe and Ni nanowires have a giant
magnetic anisotropy energy (MAE), indicating that these nanowires would have
applications in high density magnetic data storages. In this paper, we perform
density functional calculations for the Fe, Co and Ni linear atomic chains on
Cu(001) surface within the generalized gradient approximation, in order to
investigate how the substrates would affect the magnetic properties of the
nanowires. We find that Fe, Co and Ni linear chains on Cu(001) surface still
have a stable or metastable ferromagnetic state. When spin-orbit coupling (SOC)
is included, the spin magnetic moments remain almost unchanged, due to the
weakness of SOC in 3$d$ TM chains, whilst significant orbital magnetic moments
appear and also are direction-dependent. Finally, we find that the MAE for Fe,
and Co remains large, i.e., being not much affected by the presence of Cu
substrate.
