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In this paper we present an explicit construction for the fundamentalsolution to the Dirac and Laplace operator on some non-orientable conformallyflat manifolds. We first treat a class of projective cylinders and tori wherewe can study monogenic sections with values in different pin bundles. Then wediscuss the M\"obius strip, the Klein bottle and higher dimensionalgeneralizations of them. We present integral representation formulas togetherwith some elementary tools of harmonic analysis on these manifolds.

Dirac and Laplace operators on some non-orientable conformally flat manifolds