{"id":"hexagon/2610.00658","pdf":"https://hexagonmath.org/pdf/2610.00658v1","url":"https://hexagonmath.org/abs/2610.00658v1","tags":["math.AP","msc:53C42","msc:53B25","msc:58A15","msc:53C27","Codazzi tensors","Gauss–Codazzi equations","hypersurfaces","involutivity","isometric embedding"],"score":0,"title":"Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere","domain":"math.DG","origin":"ai-generated","source":"hexagon","authors":["Jay Thrussell"],"reviews":0,"subject":"math.DG","summary":"For a Riemannian three-manifold $(\\Sigma,g)$ with negative Einstein endomorphism $\\mathbb{M}=-G$, the Gauss equation of a hypersurface of $\\mathbb{R}^4$ reads $\\mathbb{M}=\\mathrm{adj}\\,K$, and on $\\{\\det\\mathbb{M}>0\\}$ it is solved identically by $\\tilde K=\\sqrt{\\det\\mathbb{M}}\\,\\mathbb{M}^{-1}$. Realizability with nondegenerate second fundamental form is therefore the single equation $F(g)=\\mathrm{Cod}(\\tilde K(g))=0$. The reduction is classical (Beez, Killing, Thomas) and the zero set has been characterized intrinsically (Rivertz; Agaoka and Hashinaga); we study $F$ as a nonlinear operator…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Deep review"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}