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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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