  Schwinger's quantization scheme is extended in order to solve the problem of
the formulation of quantum mechanics on a space with a group structure. The
importance of Killing vectors in a quantization scheme is showed. Usage of
these vectors provides algebraic properties of operators to be consistent with
the geometrical structure of a manifold. The procedure of the definition of the
quantum Lagrangian of a free particle and the norm of velocity (momentum)
operators is given. These constructions are invariant under a general
coordinate transformation. The unified procedure for constructing the quantum
theory on a space with a group structure is developed. Using it quantum
mechanics on a Riemannian manifold with a simply transitive group acting on it
is investigated.
