  We study the effect of lower dimensional geometry on the frequencies of the
collective oscillations of a Bose-Einstein condensate confined in a trap. To
study the effect of two dimensional geometry we consider a pancake-shaped
condensate confined in a harmonic trap and employ various models for the
coupling constant depending on the thickness of the condensate relative to the
the value of the scattering length. These models correspond to different
scattering regimes ranging from quasi-three dimensional to strictly two
dimensional regimes. Using these models for the coupling parameter and sum rule
approach of the many-body response theory we derive analytical expressions for
the frequencies of the monopole and the quadrupole modes. We show that the
frequencies of monopole mode of the collective oscillations are significantly
altered by the reduced dimensionality and also study the evolution of the
frequencies as the system make transition from one regime to another.
