The Kernel of the Rarita–Schwinger Operator on Riemannian Spin Manifolds

  • Yasushi Homma
  • Uwe SemmelmannEmail author


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, G2 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.


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We would like to thank Anand Dessai for several helpful comments and for his interest in our work. We also thank the anonymous referee for helpful suggestions. The first author was partially supported by JSPS KAKENHI Grant Number JP15K04858. Both authors were partially supported by the DAAD-Waseda University Partnership Programme.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of Science and EngineeringWaseda UniversityTokyoJapan
  2. 2.Institut für Geometrie und Topologie, Fachbereich MathematikUniversität StuttgartStuttgartGermany

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