  Observations suggest that our universe is spatially flat on the largest
observable scales. Exactly six different compact orientable three-dimensional
manifolds admit flat metrics. These six manifolds are therefore the most
natural choices for building cosmological models based on the present
observations. This paper briefly describes these six manifolds and the harmonic
basis functions previously developed for representing arbitrary scalar fields
on them. The principal focus of this paper is the development of new harmonics
for representing arbitrary vector and second-rank tensor fields on these
manifolds. These new harmonics are designed to be useful tools for analyzing
the dynamics of electromagnetic and gravitational fields on these spaces.
