Paper · Jul 31, 2025
Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds
As on arXiv
Abstract
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.
- Submitted
- Jul 31, 2025
- Last revised
- Jun 15, 2026 (v2)
- Subjects
- math.DG
Comments: 24 pages; v2: changed title; added one section with a second proof applying to the Riemannian case; added one section with explicit construction of pseudo-Iwasawa solvmanifolds with a Killing spinor; restated main theorem as a classification; minor corrections; presentation improved; bibliography updated
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Infinitesimal maximal symmetry and Ricci soliton solvmanifolds
Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “solvmanifolds”, “invariant Ricci”, “left invariant”
-
Inhomogeneous deformations of Einstein solvmanifolds
Topics Homogeneous spaces, Ricci solitons · Shared terms “Einstein solvmanifold”
-
Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups
Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “Iwasawa”, “nilpotent”, “Lie”
-
Solvable Lie algebras obtained by quivers and Einstein metrics
Topics Einstein metrics, Homogeneous spaces, Ricci solitons · Shared terms “Lie algebras”, “invariant Ricci”, “nilpotent Lie”
-
A Hilbert–Mumford criterion for nilsolitons
Topics Homogeneous spaces, Ricci solitons · Shared terms “nilsolitons”, “nilpotent”, “Lie algebras”
-
Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups
Topics Homogeneous spaces, Ricci solitons · Shared terms “pseudo Riemannian”, “Lie algebras”, “Lie groups”
-
Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two
Topics Homogeneous spaces, Ricci solitons · Shared terms “left invariant”, “Lie groups”, “nilsolitons”
-
Nilpotent Lie algebras obtained by quivers and Ricci solitons
Topics Homogeneous spaces, Ricci solitons · Shared terms “nilpotent Lie”, “Lie algebras”, “left invariant”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.13Homogeneous Flows
A computable laboratory: the flow on Thurston’s geometries.
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.