The importance sampling technique for stochastic integration is extended to evaluate imaginary time path integral expressions in two kinds of spacelike curved manifolds that arise frequently in the physics of constrained molecular motion. Using stereographic projection maps, we develop convenient quantum distributions. We explore the issue of energy estimation based on the extension of the virial theorem in curved manifolds and we provide simple numerical criteria to determine if the virial of a system in a curved space approaches the kinetic energy as a stochastic estimator. Simple numerical tests are carried out using both the discretized and the Fourier path integral approaches. The particle in a ring subjected to two different potentials is insightful and is sufficiently simple to simulate by other well established methods. (C) 2003 American Institute of Physics.