  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.
