  In this note we compare the spinor bundle of a Riemannian manifold
$(M=M_1\times...\times M_N,g)$ with the spinor bundles of the Riemannian
factors $(M_i,g_i)$. We show, that - without any holonomy conditions - the
spinor bundle of $(M,g)$ for a special class of metrics is isomorphic to a
bundle obtained by tensoring the spinor bundles of $(M_i,g_i)$ in an
appropriate way.
