## The Einstein-Dirac equation on Riemannian spin manifolds.(English)Zbl 0961.53023

For a Riemannian spin manifold $$(M,g)$$ we denote by $$D_g$$ the Dirae operator associated to $$g$$, by $$Ric_g$$ the Ricci-tensor and by $$S_g$$ the scalar curvature. The aim of the paper is to find exact solutions $$(g,\psi)$$ of the Riemannian Dirac-Einstein equation on a closed manifold: $$D_g\psi= \lambda \psi$$, $$Ric_g-{1\over 2}S_g \cdot g={\varepsilon \over 4}T_{(g,\psi)}$$, $$\varepsilon= \pm 1$$, $$\lambda\in \mathbb{R}$$. Special solutions of the Einstein-Dirac equation are weak Killing spinors, in dimension 3 they are the only ones. As an example special 1-connected Sasakian spin manifolds $$M^{2m+1}$$, $$m\geq 2$$, are considered. If the Ricci-tensor is related in a certain way to the contact form, then this Sasakian manifold admits weak Killing spinors. Even dimensional examples with Dirac-Einstein-spinors which are not weak Killing spinors are produced by considering products of 1-connected a nearly-Kähler manifold $$M^6$$ with manifold admitting usual geometric Killing spinors. The 3-dimensional case is examined more in detail.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C27 Spin and Spin$${}^c$$ geometry
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