  We investigate the effects of spatial curvature for an atomic Bose-Einstein
condensate confined in an elliptical waveguide. The system is well described by
an effective 1D Gross-Pitaevskii equation with a quantum-curvature potential,
which has the shape of a double-well but crucially depends on the eccentricity
of the ellipse. The ground state of the system displays a quantum phase
transition from a two-peak configuration to a one-peak configuration at a
critical attractive interaction strength. In correspondence of this phase
transition the superfluid fraction strongly reduces and goes to zero for a
sufficiently attractive Bose-Bose interaction.
