Extended Schwinger's quantization procedure is used for constructing quantum
mechanics on a manifold with a group structure. The considered manifold M is
a homogeneous Riemannian space with the given action of isometry transformation
group. Using the identification of M with the quotient space G/H, where H
is the isotropy group of an arbitrary fixed point of M, we show that quantum
mechanics on G/H possesses a gauge structure, described by the gauge
potential that is the connection 1-form of the principal fiber bundle G(G/H,H). The coordinate representation of quantum mechanics and the procedure for
selecting the physical sector of states are developed.Comment: 18pages, no figures, LaTe