On pseudo-Riemannian manifolds with many Killing spinors

TL;DR

This paper establishes conditions under which pseudo-Riemannian manifolds with many Killing or conformal Killing spinors are locally homogeneous or have special geometric properties, extending known results and providing classifications.

Contribution

It proves new bounds on the number of Killing spinors that imply local homogeneity or special curvature properties, and offers a classification of Riemannian manifolds with Killing spinors.

Findings

Manifolds with more than 3/4 N Killing spinors are locally homogeneous.

More than 3/4 N parallel spinors imply the manifold is flat.

Classification results for Riemannian manifolds with 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.