An odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with real Killing constant α if and only if it is α-Sasakian.
Friedrich,Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnega- tiver Skalarkrümmung, Math
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Sasakian manifolds and spin-c Killing spinors
An odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with real Killing constant α if and only if it is α-Sasakian.