  We investigate the effects of dimensional reduction in atomic Bose-Einstein
condensates (BECs) induced by a strong harmonic confinement in the cylindric
radial direction or in the cylindric axial direction. The former case
corresponds to a transition from 3D to 1D in cigar-shaped BECs, while the
latter case corresponds to a transition from 3D to 2D in disc-shaped BECs. We
analyze the first sound velocity in axially-homogeneous cigar-shaped BECs and
in radially-homogeneous disc-shaped BECs. We consider also the dimensional
reduction in a BEC confined by a harmonic potential both in the radial
direction and in the axial direction. By using a variational approach, we
calculate monopole and quadrupole collective oscillations of the BEC. We find
that the frequencies of these collective oscillations are related to the
dimensionality and to the repulsive or attractive inter-atomic interaction.
