  Motivated by the recent achievement of space-based Bose-Einstein condensates
(BEC) with ultracold alkali-metal atoms under microgravity and by the proposal
of bubble traps which confine atoms on a thin shell, we investigate the BEC
thermodynamics on the surface of a sphere. We determine analytically the
critical temperature and the condensate fraction of a noninteracting Bose gas.
Then we consider the inclusion of a zero-range interatomic potential, extending
the noninteracting results at zero and finite temperature. Both in the
noninteracting and interacting cases the crucial role of the finite radius of
the sphere is emphasized, showing that in the limit of infinite radius one
recovers the familiar two-dimensional results. We also investigate the
Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on
the surface of the sphere, analyzing the interplay of condensation and
superfluidity in this finite-size system.
