Conformally covariant nonlinear equations on tensor-spinors

Abstract

We prove that to each conformally covariant equation on tensor-spinors on a Riemannian or pseudo-Riemannian manifold with spin structure one can add a nonlinear term without losing the property of conformal covariance. It follows in particular that, on a manifold of dimension n, the nonlinear Dirac equation, Pψ + λ|ψ|1/(n−1)ψ = 0, where P is the Dirac operator and λ is a constant, is conformally covariant. This generalizes a result of Gursey [1]. Some results of Orsted [2], concerning a nonlinear equation associated with the Laplacian on function, and of Branson, concerning distinguished nonlinearities associated with his modified Laplacian on differential forms [3] are also derived as particular cases of this general result.