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

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