The Cauchy problem for metrics with parallel spinors

We show that in the analytic category, given a Riemannian metric g on a hypersurfaceM � Z and a symmetric tensorW onM, the metric g can be locally extended to a Riemannian Einstein metric on Z with second fundamental form W, provided that g and W satisfy the constraints on M imposed by the contracted Codazzi equations. We use this fact to study the Cauchy problem for metrics with parallel spinors in the real analytic category and give an a�rmative answer to a question raised in [15]. We also answer negatively the corresponding questions in the smooth category