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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 $M$ islocally homogeneous if it admits more than ${3-4}N$ independent Killing spinorswith the same Killing number, unless $n\equiv 1 \pmod 4$ and $s\equiv 3 \pmod4$. We also prove that $M$ is locally homogeneous if it admits $k +$independent Killing spinors with Killing number $\lambda$ and $k -$ independentKilling 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 withmore than ${3-4}N$ independent \emph{conformal} Killing spinors is\emph{conformally} locally homogeneous. For positive or negative definitemetrics, the bounds ${3-4}N$ and ${3-2}N$ in the above results can be relaxedto ${1-2}N$ and $N$, respectively. Furthermore, we prove that apseudo-Riemannnian spin manifold with more than ${3-4}N$ parallel spinors isflat 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 spinorswith 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 completeRiemannian manifolds admitting Killing spinors, which provides an inductiveconstruction of such manifolds.



Author: D.V. Alekseevsky, V. Cort├ęs

Source: https://arxiv.org/







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