Nucleation of bended vortices in Bose-Einstein condensates in elongated traps

We study the generation of vortices in rotating axially elongated magneto-optical traps, a situation which has been realized in a recent experiment (K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)). We predict that at a critical frequency the condensate experiences a symmetry breaking and changes from a convex cloud to a state with one bended vortex. We also discuss several effects which enlarge the critical frequency with respect to other geometries of the trap: these are, (i) the failure of the Thomas-Fermi approximation on the transverse degrees of freedom of the condensate, (ii) the enhancement of the transverse asymmetry of the trap by means of rotation and (iii) the yet unobserved bending of the vortex lines.