The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion K\"ahler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.