Notes on Feynman path integral-like methods of quantization on Riemannian manifolds

Abstract: We propose an alternative method for Feynman path integrals on compact Riemannian manifolds. Our method employs action integrals along the shortest paths. In the case of rank 1 locally symmetric Riemannian manifolds, we prove the strong convergence of time slicing products of oscillatory integrals for low energy functions. Moreover, the strong limit includes Dewitt curvature $R/6$, where $R$ denotes the scalar curvature of a Riemannian manifold.