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In this work we consider a suitable generalization of the Feynman pathintegral on a specific class of Riemannian manifolds consisting of compact Liegroups with bi-invariant Riemannian metrics. The main tools we use are theCartan development map, the notion of oscillatory integral, and the Chernoffapproximation theorem. We prove that, for a class of functions of a densesubspace of the relevant Hilbert space, the Feynman map produces the solutionof the Schr\"odinger equation, where the Laplace-Beltrami operator coincideswith the second order Casimir operator of the group.

Feynman path integrals on compact Lie groups with bi-invariant Riemannian metrics and Schr\"odinger equations