Cotangent type functions in $\mathbb{R}^n$ are used to construct
Cauchy kernels and Green kernels on the conformally flat manifolds
$\mathbb{R}^n / \mathbb{Z}^k$ where $1\leq k\leq n$. Basic
properties of these kernels are discussed including introducing a
Cauchy formula, Green's formula, Cauchy transform, Poisson kernel,
Szegö kernel and Bergman kernel for certain types of domains.
Singular Cauchy integrals are also introduced as are associated
Plemelj projection operators. These in turn are used to study Hardy
spaces in this context. Also the analogues of Calderón-Zygmund type
operators are introduced in this context, together with singular
Clifford holomorphic, or monogenic, kernels defined on sector
domains in the context of cylinders. Fundamental differences in the
context of the $n$-torus arising from a double singularity for the
generalized Cauchy kernel on the torus are also discussed.