  The random walk of a bonded monomer in a polymer melt is anisotropic due to
local order and bond connectivity. We investigate both effects by
molecular-dynamics simulations on melts of fully-flexible linear chains ranging
from dimers ($M=2$) up to entangled polymers ($M=200$). The corresponding
atomic liquid is also considered as reference system. To disentangle the
influence of the local geometry and the bond arrangements, and reveal their
interplay, we define suitable measures of the anisotropy emphasising either the
former or the latter aspect. Connectivity anisotropy, as measured by the
correlation between the initial bond orientation and the direction of the
subsequent monomer displacement, shows a slight enhancement due to the local
order at times shorter than the structural relaxation time. At intermediate
times - when the monomer displacement is comparable to the bond length - a
pronounced peak and then decays slowly as $t^{-1/2}$, becoming negligible when
the displacements is as large as about five bond lengths, i.e. about four
monomer diameters or three Kuhn lengths. Local-geometry anisotropy, as measured
by the correlation between the initial orientation of a characteristic axis of
the Voronoi cell and the subsequent monomer dynamics, is affected at shorter
times than the structural relaxation time by the cage shape with antagonistic
disturbance by the connectivity. Differently, at longer times, the connectivity
favours the persistence of the local-geometry anisotropy which vanishes when
the monomer displacement exceeds the bond length. Our results strongly suggest
that the sole consideration of the local order is not enough to understand the
microscopic origin of the rattling amplitude of the trapped monomer in the cage
of the neighbours.
