Let M be a pseudo‐Riemannian spin manifold of dimension n and signature s and denote by N the rank of the real spinor bundle. We prove that M is locally homogeneous if it admits more than independent Killing spinors with the same Killing number, unless (mod 4) and (mod 4). We also prove that M is locally homogeneous if it admits independent Killing spinors with Killing number λ and independent Killing spinors with Killing number such that unless (mod 4). Similarly, a pseudo‐Riemannian manifold with more than independent conformal Killing spinors is conformally locally homogeneous. For (positive or negative) definite metrics, the bounds and in the above results can be relaxed to and N, respectively. Furthermore, we prove that a pseudo‐Riemannnian spin manifold with more than parallel spinors is flat and that parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than Killing spinors with the Killing number has constant curvature For Lorentzian or negative definite metrics the same is true with the bound Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.
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Research Article| February 02 2009
On pseudo‐Riemannian manifolds with many Killing spinors
D. V. Alekseevsky;
AIP Conf. Proc. 1093, 3–18 (2009)
D. V. Alekseevsky, V. Cortés; On pseudo‐Riemannian manifolds with many Killing spinors. AIP Conf. Proc. 2 February 2009; 1093 (1): 3–18. https://doi.org/10.1063/1.3089206
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