Harmful structures and Killing spinors on unimodular Lie groups

Abstract

A pseudo-Riemannian Einstein manifold with a Killing spinor and Killing constant $\hat{I}»$ induces on its nondegenerate hypersurfaces a pair of spinors $\ddot{I}•,\ddot{I}ˆ$ and a symmetric tensor $A$, corresponding to the second fundamental form. Viewed as an intrinsic object, $(\ddot{I}•,\ddot{I}ˆ,A,\hat{I}»)$ is known as a harmful structure; this notion generalizes nearly hypo and nearly half-flat structures to arbitrary dimension and signature. We show that when $A$ is a multiple of the identity the harmful structure is determined by a Killing spinor. We characterize left-invariant harmful structures on Lie groups in terms of Clifford multiplication by some special elements induced by the structure constants and metric. This enables us to classify left-invariant harmful structures on unimodular metric Lie groups of definite or Lorentzian signature and dimension $\le 4$, under the assumption that the symmetric tensor $A$ is diagonalizable over $\mathbb{R}$. These pseudo-Riemannian Lie groups are principal orbits of cohomogeneity one Einstein metrics of Riemannian, Lorentzian or anti-Lorentzian signature with a Killing spinor.

58 pages, 4 tables; v2: 59 pages, corrected Proposition 8.3 (Lorentzian case) with a reparameterization, plus minor clarifications and typo fixes