{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/indefinite-nilsolitons-and-einstein","title":"Indefinite nilsolitons and Einstein solvmanifolds","arxiv_id":"2105.09209","date":"2021-05-19","proceeding":null,"authors":["Diego Conti","Federico A. Rossi"],"abstract":"A nilsoliton is a nilpotent Lie algebra $\\mathfrak{g}$ with a metric such that $\\operatorname{Ric}=\\lambda \\operatorname{Id}+D$, with $D$ a derivation. For indefinite metrics, this determines four different geometries, according to whether $\\lambda$ and $D$ are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that $D$ must satisfy when it is nonzero. For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold of the form $\\mathfrak{g}\\rtimes \\mathbb{R}^k$. Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form $\\mathfrak{g}\\rtimes \\mathbb{R}^k$ that determine a nilsoliton metric on $\\mathfrak{g}$ by restriction. We show with examples that, unlike in the Riemannian case, one cannot establish a correspondence between the full classes of Einstein solvmanifolds and nilsolitons.","url_abs":"https://arxiv.org/abs/2105.09209v2","url_pdf":"https://arxiv.org/pdf/2105.09209v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"indefinite-nilsolitons-and-einstein","repo_url":"https://github.com/diego-conti/gleipnir","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}