On pseudo-Riemannian manifolds with many Killing spinors

Abstract

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. For (positive or negative) definite metrics, the bounds ${3/4}N$ and ${3/2}N$ in the above results can be relaxed to ${1/2}N$ and $N$, respectively. Furthermore, we prove that a pseudo-Riemannnian spin manifold with more than ${3/4}N$ parallel spinors is flat and that ${1/4}N$ parallel spinors suffice if the metric is definite. Similarly, a Riemannnian spin manifold with more than ${3/8}N$ Killing spinors with the Killing number $\lambda \in \bR$ has constant curvature $4\lambda^2$. For Lorentzian or negative definite metrics the same is true with the bound ${1/2}N$. Finally, we give a classification of (not necessarily complete) Riemannian manifolds admitting Killing spinors, which provides an inductive construction of such manifolds.