The Einstein-Dirac equation on Sasakian 3-manifolds

We prove that a Sasakian 3-manifold admitting a non-trivial solution to the Einstein-Dirac equation has necessarily constant scalar curvature. In the case when this scalar curvature is non-zero, their classification follows then from a result by Th. Friedrich and E.C. Kim. We also prove that a scalar-flat Sasakian 3-manifold admits no local Einstein spinors. (C) 2001 Elsevier Science B.V. All rights reserved.