  In this work we consider a suitable generalization of the Feynman path
integral on a specific class of Riemannian manifolds consisting of compact Lie
groups with bi-invariant Riemannian metrics. The main tools we use are the
Cartan development map, the notion of oscillatory integral, and the Chernoff
approximation theorem. We prove that, for a class of functions of a dense
subspace of the relevant Hilbert space, the Feynman map produces the solution
of the Schr\"odinger equation, where the Laplace-Beltrami operator coincides
with the second order Casimir operator of the group.
