Warped Product Einstein Manifolds in Four Dimensions

[2606.08047] Warped Product Einstein Manifolds in Four Dimensions Skip to main content Search Submit Donate Log in General Relativity and Quantum Cosmology arXiv:2606.08047 (gr-qc) [Submitted on 6 Jun 2026] Title: Warped Product Einstein Manifolds in Four Dimensions Authors: Jack C. M. Hughes , Joudy F. Jamal Beek , Fedor V. Kusmartsev View PDF HTML (experimental) Abstract: On four-dimensional (pseudo-)Riemannian manifolds $\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \times 6$ matrix representation which decomposes into four $3 \times 3$ blocks. $\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\mathcal{M} = F \times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity transformation can be found between them. We construct these matrices and associated transformations for $1+3$, $2+2$ and $3+1$ warped products, giving classifications for the Einstein limits from this algebraic perspective. Using this, one can easily Petrov classify the Einstein warped products for each case considered: $3+1$ are generically type-\RNum{1}, $2+2$ are type-D while $1+3$ are constrained to be type-O. In the closed Riemannian case, there are a number of topological restrictions on these manifolds that we discuss: in the half-conformally flat limit, each of these Einstein warped products must be flat. Subjects: General Relativity and Quantum Cosmology (gr-qc) Cite as: arXiv:2606.08047 [gr-qc] (or arXiv:2606.08047v1 [gr-qc] for this version) https://doi.org/10.48550/arXiv.2606.08047 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Journal of Geometry and Physics 230 (2026) 105987 Related DOI : https://doi.org/10.1016/j.geomphys.2026.105987 Focus to learn more DOI(s) linking to related resources Submission history From: Jack Hughes [ view email ] [v1] Sat, 6 Jun 2026 08:20:56 UTC (330 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: gr-qc < prev | next > new | recent | 2026-06 References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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