  Different approaches are compared to formulation of quantum mechanics of a
particle on the curved spaces. At first, the canonical, quasi-classical and
path integration formalisms are considered for quantization of geodesic motion
on the Rimannian configurational spaces. A unique rule of ordering of operators
in the canonical formalism and a unique definition of the path integral are
established and, thus, a part of ambiguities in the quantum counterpart of
geodesic motion is removed. A geometric interpretation is proposed for
non-invariance of the quantum mechanics on coordinate transformations. An
approach alternative to the quantization of geodesic motion is surveyed, which
starts with the quantum theory of a neutral scalar field. Consequences of this
alternative approach and the three formalisms of quantization are compared. In
particular, the field theoretical approach generates a deformation of the
canonical commutation relations between coordinates and momenta of a prticle. A
possible cosmological consequence of the deformation is presented in short. Key
words: quantum mechanics, Riemannian space, geodesic motion, deformation.
