  In this paper we use the G-spin theorem to show that the Davis hyperbolic
4-manifold admits harmonic spinors. This is the first example of a closed
hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe
the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to
calculated the subtle sign terms in the G-spin theorem for an isometry, with
isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.
