On pseudo-Riemannian manifolds with many Killing spinors

Understand

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 ${3/4}N$ independent Killing spinors with the same Killing number, unless $n\equiv 1 \pmod 4$ and $s\equiv 3 \pmod 4$.
• We also prove that $M$ is locally homogeneous if it admits $k_+$ independent Killing spinors with Killing number $\lambda$ and $k_-$ independent Killing spinors with Killing number $-\lambda$ such that $k_++k_->{3/2}N$, unless $n\equiv s\equiv 3\pmod 4$.
• Similarly, a pseudo-Riemannian manifold with more than ${3/4}N$ independent \emph{conformal} Killing spinors is \emph{conformally} locally homogeneous.