Real Killing spinors and holonomy
The author describes all complete simply connected Riemannian manifolds carrying real Killing spinors. All these manifolds are automatically compact. The results are roughly as follows: (1) In any even dimension \(n\neq 6\) only the standard spheres carry real Killing spinors. In dimension 6 one has in addition the nearly Kähler, non-Kähler manifolds. (2) For \(n=4m+1\), \(m\geq 1\) only the standard sphere and the Einstein-Sasaki manifolds carry real Killing spinors. (3) For \(n=4m+3\), \(m\geq 2\) one gets the standard sphere, the Einstein-Sasaki manifolds, and the Sasaki-3-manifolds. In the remaining dimension 7 there are four possibilities (for details we refer to the paper). As a byproduct, the exceptional cases \(n=6,7\) yield construction methods for Riemannian manifolds with the exceptional holonomy groups \(G_ 2\) (in dimension 6) and Spin(7) (in dimension 7).
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