  The phenomenon of rotation of a vector under parallel transport along a
closed path is known as anholonomy. In this paper we have studied the
anholonomy for noncontractible loops - closed paths in a curved surface that do
not enclose any area and hence Stokes theorem is not directly applicable.
Examples of such closed paths are poloidal and toroidal loops and knots on a
torus. The present study is distinct from conventional results on anholonomy
for closed paths on $S_2$ since in the latter case all closed paths are
contractible or trivial cycles. We find that for some nontrivial cycles the
anholonomy cancels out over the complete cycle. Next we calculate Hannay angle
for a particle traversing such noncontractible loops when the torus itself is
revolving. Some new and interesting results are obtained especially for
poloidal paths that is for paths that encircle the torus ring.
