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 $3\mathrm{/}4N$ independent Killing spinors with the same Killing number, unless $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $s\equiv 3 \pmod 4$\mathrm{.}Wealsoprovethat$M$islocallyhomogeneousifitadmits$k_+$independentKillingspinorswithKillingnumber$\lambda$and$k_-$ independent Killing spinors with Killing number $-\lambda$ such that $k_{+}+k_{-}>3\mathrm{/}2N$, unless $n\equiv s\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Similarly, a pseudo-Riemannian manifold with more than $3\mathrm{/}4N$ independent \emph{conformal} Killing spinors is \emph{conformally} locally homogeneous. For (positive or negative) definite metrics, the bounds $3\mathrm{/}4N$ and $3\mathrm{/}2N$ in the above results can be relaxed to $1\mathrm{/}2N$ and $N$, respectively. Furthermore, we prove that a pseudo-Riemannnian spin manifold with more than $3\mathrm{/}4N$ parallel spinors is flat and that $1\mathrm{/}4N$ parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than $3\mathrm{/}8N$ Killing spinors with the Killing number $\lambda \in \text{bR}$ has constant curvature $4{\lambda }^2$. For Lorentzian or negative definite metrics the same is true with the bound $1\mathrm{/}2N$. Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.