  This paper focuses on the development of harmonic and Clifford analysis
techniques in the context of some conformally flat manifolds that arise from
factoring out a simply-connected domain from $R^n$ by special arithmetic
subgroups of the conformal group. Our discussion encompasses in particular the
Hopf manifold $S^1 \times S^{n-1}$, conformally flat cylinders and tori and
some conformally flat manifolds of genus $g \ge 2$, such as $k$-handled tori
and polycylinders. This paper provides a continuation as well as an extension
of our previous two papers \cite{KraRyan1,KraRyan2}. In particular, we
introduce a Cauchy integral formula for hypermonogenic functions on cylinders,
tori and on half of the Hopf manifold. These are solutions to the Dirac-Hodge
equation with respect to the hyperbolic metric. We further develop
generalizations of the Mittag-Leffler theorem and the Laurent expansion theorem
for cylindrical and toroidal monogenic functions. The study of Hardy space
decompositions on the Hopf manifold is also continued. Kerzman-Stein operators
are introduced. Explicit formulas for the Szeg\"o kernel, the Bergman kernel
and the Poisson kernel of half the Hopf manifold are given.
