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Real Killing spinors and holonomy. (English) Zbl 0778.53037
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).

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C20 Global Riemannian geometry, including pinching
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