  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.
