  Relativistic Riemannian superfluid hydrodynamics used in general relativity
to investigate superfluids in pulsars is extended to non-Riemannian background
spacetime endowed with Cartan torsion. From the Gross-Pitaeviskii (GP) it is
shown that in the weak field Cartan torsion approximation, the torsion vector
is orthogonal to the superfluid plane wave velocity. Torsion vector is also
shown to be aligned along the vortex direction in the superfluid. The
background torsion is shown to induce rotation on the fluid as happens with the
acoustic torsion in the analogue non-Riemannian non-relativistic superfluid
models. The torsion part of the current would be connected to the normal part
of the superfluid velocity while the Riemannian part of the velocity would be
connected to the superfluid velocity itself. Magnus effect and the rotation of
the superfluid are analysed. Since the Kalb-Ramond field is easily associated
with torsion our method seems to be equivalent to the vortex-cosmic string
relativistic superfluid method developed by Carter and Langlois to investigate
rotating neutron stars.
