{"paper":{"title":"Four dimensional static and related critical spaces with harmonic curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jinwoo Shin, Jongsu Kim","submitted_at":"2016-04-12T04:12:55Z","abstract_excerpt":"In this article we study any 4-dimensional Riemannian manifold $(M,g)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the following equation \\begin{eqnarray} \\label{0002bx} \\nabla df = f(Rc -\\frac{R}{n-1} g) + x Rc+ y(R) g. \\end{eqnarray} where $Rc$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. We show that a neighborhood of any point in some open dense subset of $M$ is locally isometric to one of the following five types; {\\rm (i)} $ \\mathbb{S}^2(\\frac{R}{6}) \\times \\mathbb{S}^2(\\frac{R}{3})$ with $R>0$, {\\rm (ii)} $ \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.03241","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}