  We construct exact solutions of the Einstein-Dirac equation, which couples
the gravitational field with an eigenspinor of the Dirac operator via the
energy-momentum tensor. For this purpose we introduce a new field equation
generalizing the notion of Killing spinors. The solutions of this spinorial
field equation are called weak Killing spinors (WK-spinors). They are special
solutions of the Einstein-Dirac equation and in dimension n=3 the two equations
essentially coincide. It turns out that any Sasakian manifold with Ricci tensor
related in some special way to the metric tensor as well as to the contact
structure admits a WK-spinor. This result is a consequence of the investigation
of special spinorial field equations on Sasakian manifolds (Sasakian
quasi-Killing spinors). Altogether, in odd dimensions a contact geometry
generates a solution of the Einstein-Dirac equation. Moreover, we prove the
existence of solutions of the Einstein-Dirac equations that are not WK-spinors
in all dimensions n > 8.
