In this article, we give all the Weitzenböck-type formulas among the geometric first-order differential operators on the spinor fields with spin j+ 1 / 2 over Riemannian spin manifolds of constant curvature. Then, we find an explicit factorization formula of the Laplace operator raised to the power j+ 1 and understand how the spinor fields with spin j+ 1 / 2 are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge–de Rham decomposition on spaces of constant curvature.
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