arXiv subjects

Romeo Segnan Dalmasso

Publications and source records attributed to Romeo Segnan Dalmasso.

7 recordsLinked to original sources

arXiv2026

Harmful structures and Killing spinors on unimodular Lie groups

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.

math.DG↗

arXiv2026

Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds

Riemannian Einstein solvmanifolds can be described in terms of nilsolitons, namely nilpotent Lie groups endowed with a left-invariant Ricci soliton metric. This characterization does not extend to indefinite metrics; nonetheless, nilsolitons can be defined and used to construct Einstein solvmanifolds of a higher dimension in any signature. An Einstein solvmanifold obtained by this construction turns out to satisfy the pseudo-Iwasawa condition, meaning that its Lie algebra splits as the orthogonal sum of a nilpotent ideal and an abelian subalgebra, the latter acting by symmetric derivations. In this paper we construct a family of pseudo-Iwasawa solvmanifolds admitting a Killing spinor in any dimension and signature and prove that all pseudo-Iwasawa solvmanifolds admitting a Killing spinor, invariant or not, belong to this family. If in addition the metric is Einstein, we show that the only possibility is the hyperbolic half-space. As a byproduct, we prove that the only homogeneous Riemannian manifold admitting a Killing spinor with imaginary Killing constant is hyperbolic space.

math.DG↗

arXiv2024

A construction of Einstein solvmanifolds not based on nilsolitons

We construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form $\mathfrak{g}\rtimes_D\mathbb{R}$, where $\mathfrak{g}$ is a nilpotent Lie algebra and $D$ is a nonsymmetric derivation. Considering nonsymmetric derivations has the consequence that $\mathfrak{g}$ is not a nilsoliton, but satisfies a more general condition. Our construction is based on the notion of nondiagonal triple on a nice diagram. We present an algorithm to classify nondiagonal triples and the associated Einstein metrics. With the use of a computer, we obtain all solutions up to dimension $5$, and all solutions in dimension $\leq9$ that satisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension $\leq 5$ that we obtain by our construction are not isometric to a standard extension of a nilsoliton.

math.DG↗

arXiv2024

Killing spinors and hypersurfaces

We consider spin manifolds with an Einstein metric, either Riemannian or indefinite, for which there exists a Killing spinor. We describe the intrinsic geometry of nondegenerate hypersurfaces in terms of a PDE satisfied by a pair of induced spinors, akin to the generalized Killing spinor equation. Conversely, we prove an embedding result for real analytic pseudo-Riemannian manifolds carrying a pair of spinors satisfying this condition.

math.DG↗

arXiv2023

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of $\mathfrak z$-standard Sasaki solvable Lie algebras of dimension $2n+3$, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension $2n$ endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics. We classify $\mathfrak z$-standard Sasaki solvable Lie algebras of dimension $\leq 7$ and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

math.DG↗

arXiv2023

Pseudo-Riemannian Sasaki solvmanifolds

We study a class of left-invariant pseudo-Riemannian Sasaki metrics on solvable Lie groups, which can be characterized by the property that the zero level set of the moment map relative to the action of some one-parameter subgroup $\{\exp tX\}$ is a normal nilpotent subgroup commuting with $\{\exp tX\}$, and $X$ is not lightlike. We characterize this geometry in terms of the Sasaki reduction and its pseudo-Kähler quotient under the action generated by the Reeb vector field. We classify pseudo-Riemannian Sasaki solvmanifolds of this type in dimension $5$ and those of dimension $7$ whose Kähler reduction in the above sense is abelian.

math.DG↗

arXiv2022

Diagonalization of the metric of a Lorentzian 3-manifold

We study the problem of diagonalization of the metric of 3-dimensional Lorentzian manifold. Applying the technique of moving frames, we prove that every smooth Lorentzian 3-manifold admits an atlas in which the metric assumes a diagonal form.

math.DG↗