Harmful structures and Killing spinors on unimodular Lie groups
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| Format: | Preprint |
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2025
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| author | Conti, Diego Rossi, Federico A. Dalmasso, Romeo Segnan |
| author_facet | Conti, Diego Rossi, Federico A. Dalmasso, Romeo Segnan |
| contents | A pseudo-Riemannian Einstein manifold with a Killing spinor and Killing constant $λ$ induces on its nondegenerate hypersurfaces a pair of spinors $ϕ,ψ$ and a symmetric tensor $A$, corresponding to the second fundamental form. Viewed as an intrinsic object, $(ϕ,ψ,A,λ)$ 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 $\leq 4$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmful structures and Killing spinors on unimodular Lie groups Conti, Diego Rossi, Federico A. Dalmasso, Romeo Segnan Differential Geometry 53C25 (Primary) 53C50, 53C27, 53C30, 22E60 (Secondary) A pseudo-Riemannian Einstein manifold with a Killing spinor and Killing constant $λ$ induces on its nondegenerate hypersurfaces a pair of spinors $ϕ,ψ$ and a symmetric tensor $A$, corresponding to the second fundamental form. Viewed as an intrinsic object, $(ϕ,ψ,A,λ)$ 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 $\leq 4$, 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. |
| title | Harmful structures and Killing spinors on unimodular Lie groups |
| topic | Differential Geometry 53C25 (Primary) 53C50, 53C27, 53C30, 22E60 (Secondary) |
| url | https://arxiv.org/abs/2509.08507 |