Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S^1

We consider the uncertainty relation between position and momentum of a particle on $S^1$ (a circle). Since $S^1$ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $S^D$ (a $D$-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.