  We formulate path integrals on any Riemannian manifold which admits the
action of a compact Lie group by isometric transformations. We consider a path
integral on a Riemannian manifold M on which a Lie group G acts isometrically.
Then we show that the path integral on M is reduced to a family of path
integrals on a quotient space Q=M/G and that 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 our formulation is applicable to a wide class of manifolds, which
includes inhomogeneous spaces, and it covers all the inequivalent
quantizations. To describe the path integral on inhomogeneous space,
stratification geometry, which is a generalization of the concept of principal
fiber bundle, is necessarily introduced. Using it we show that the path
integral is expressed as a product of three factors; the rotational energy
amplitude, the vibrational energy amplitude, and the holonomy factor. When a
singular point arises in $ Q $, we determine the boundary condition of the path
integral kernel for a path which runs through the singularity.
