Non-Riemannian acoustic spacetime of vortex hydrodynamics in   Bose-Einstein condensates
  Applications of non-Riemannian acoustic geometries in Bose-Einstein condensates (BEC) are considered. The first is the Minkowski-Cartan irrotational vortex acoustic geometry of nonlinear Schr\"{o}dinger equations of BEC (Gross-Pitaeviskii (GP) equation). In this model, which is an alternative to the Riemannian acoustic geometry of phonons in BEC, the Cartan acoustic torsion is physically interpreted as the bending of the BEC wave function amplitude. Actually this shows that acoustic torsion is given by the density perturbation of BEC flow as happens in relativistic cosmological fluid spacetimes. The Ricci-Cartan curvature scalar is computed and a torsion singularity is found at the origin of a quantized vortex in BEC. In the second example, a transverse Magnus force is shown to be expressed in terms of acoustic torsion on a teleparallel vortex acoustics geometry. 
L.C. Garcia de Andrade