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 Kähler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita–Schwinger fields. In the case of Calabi–Yau, hyperkähler, 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.