  Motivated by numerous experiments on Bose-Einstein condensed atoms which have
been performed in tight trapping potentials of various geometries (elongated
and/or toroidal/annular), we develop a general method which allows us to reduce
the corresponding three-dimensional Gross-Pitaevskii equation for the order
parameter into an effectively one-dimensional equation, taking into account the
interactions (i.e., treating the width of the transverse profile variationally)
and the curvature of the trapping potential. As an application of our model we
consider atoms which rotate in a toroidal trapping potential. We evaluate the
state of lowest energy for a fixed value of the angular momentum within various
approximations of the effectively one-dimensional model and compare our results
with the full solution of the three-dimensional problem, thus getting evidence
for the accuracy of our model.

Quantum Gases;Pattern Formation and Solitons;Atomic Physics;Quantum Physics