On pseudo‐Riemannian manifolds with many Killing spinors

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 $34N$ independent Killing spinors with the same Killing number, unless $n ≡ 1$ (mod 4) and $s ≡ 3$ (mod 4). We also prove that M is locally homogeneous if it admits $k+$ independent Killing spinors with Killing number λ and $k−$ independent Killing spinors with Killing number $−λ$ such that $k++k−>32N,$ unless $n ≡ s ≡ 3$ (mod 4). Similarly, a pseudo‐Riemannian manifold with more than $34N$ independent conformal Killing spinors is conformally locally homogeneous. For (positive or negative) definite metrics, the bounds $34N$ and $32N$ in the above results can be relaxed to $12N$ and N, respectively. Furthermore, we prove that a pseudo‐Riemannnian spin manifold with more than $34N$ parallel spinors is flat and that $14N$ parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than $38N$ Killing spinors with the Killing number $λ∈R$ has constant curvature $4λ2.$ For Lorentzian or negative definite metrics the same is true with the bound $12N.$ Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.