  We investigate the effects of dimensional reduction in Bose gases induced by
a strong harmonic confinement in the transverse cylindric radial direction. By
using a generalized Lieb-Liniger theory, based on a variational treatment of
the transverse width of the Bose gas, we analyze the transition from a 3D
Bose-Einstein condensate to the 1D Tonks-Girardeau gas. The sound velocity and
the frequency of the lowest compressional mode give a clear signature of the
regime involved. We study also the case of negative scattering length deriving
the phase diagram of the Bose gas (uniform, single soliton, multi soliton and
collapsed) in toroidal confinement.
