Parallel spinors on flat manifolds

Abstract

Let p ( M ) be the dimension of the vector space of parallel spinors on a closed spin manifold M. We prove that every finite group G is the holonomy group of a closed flat spin manifold M ( G ) such that p ( M ( G ) ) > 0 . If the holonomy group Hol ( M ) of M is cyclic, then we give an explicit formula for p ( M ) another than that given in [R.J. Miatello, R.A. Podesta, The spectrum of twisted Dirac operators on compact flat manifolds, Trans. Am. Math. Soc., in press]. We answer the question when p ( M ) > 0 if Hol ( M ) is a cyclic group of prime order or dim ⁡ M ≤ 4 .