A Local Existence Theorem for the Einstein-Dirac Equation

We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold $M^n$ admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics $\bar{\eta}$ on $M^n \times {\mathbb R}$ such that $(M^n \times {\mathbb R}, \bar{\eta})$ admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints.