  We study a quantum system in a Riemannian manifold M on which a Lie group G
acts isometrically. The path integral on M is decomposed into a family of path
integrals on a quotient space Q=M/G and the reduced path integrals are
completely classified by irreducible unitary representations of G. It is not
necessary to assume that the action of G on M is either free or transitive.
Hence the quotient space M/G may have orbifold singularities. Stratification
geometry, which is a generalization of the concept of principal fiber bundle,
is necessarily introduced to describe the path integral on M/G. Using it we
show that the reduced path integral is expressed as a product of three factors;
the rotational energy amplitude, the vibrational energy amplitude, and the
holonomy factor.
