  The hyperspherical adiabatic method is used to derive stability criteria for
Bose-Einstein condensates in deformed external fields. An analytical
approximation is obtained. For constant volume the highest stability is found
for spherical traps. Analytical approximations to the stability criterion with
and without zero point motion are derived. Extreme geometries of the field
effectively confine the system to dimensions lower than three. As a function of
deformation we compute the dimension to vary continuously between one and
three. We derive a dimension-dependent effective radial Hamiltonian and
investigate one choice of an effective interaction in the deformed case.
