Integrals over Products of Distributions from Manifest Coordinate Invariance of Perturbation Expansions of Path Integrals in Curved Space

We show that the requirement of manifest coordinate invariance of perturbatively defined quantum-mechanical path integrals in curved space leads to an extension of the theory of distributions by specifying unique rules for integrating products of distributions. The rules are derived using equations of motion and partial integration, while keeping track of certain minimal features stemming from the unique definition of all singular integrals in 1 - epsilon dimensions. Our rules guarantee complete agreement with much more cumbersome calculations in 1- epsilon dimensions where the limit epsilon --> 0 is taken at the end. In contrast to our previous papers where we solved the same problem for an infinite time interval or zero temperature, we consider here the more involved case of finite-time (or non-zero temperature) amplitudes.