  The Bose-Einstein condensates of alkali atomic gases are spinor fields with
local ``spin-gauge" symmetry. This symmetry is manifested by a superfluid
velocity ${\bf u}_{s}$ (or gauge field) generated by the Berry phase of the
spin field. In ``static" traps, ${\bf u}_{s}$ splits the degeneracy of the
harmonic energy levels, breaks the inversion symmetry of the vortex nucleation
frequency ${\bf \Omega}_{c1}$, and can lead to {\em vortex ground states}. The
inversion symmetry of ${\bf \Omega}_{c1}$, however, is not broken in ``dynamic"
traps. Rotations of the atom cloud can be generated by adiabatic effects
without physically rotating the entire trap.
