  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.
