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Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere

Primarily AI-generated textHuman understanding: no partsmath.DG — Differential Geometrymath.AP — Analysis of PDEs

Contributed by Jay Thrussell

Version 2 / Oct 29, 2026 / CC BY 4.0

Abstract

For a Riemannian three-manifold (Σ,g)(\Sigma,g) with negative Einstein endomorphism M=−G\mathbb{M}=-G, the Gauss equation of a hypersurface of R4\mathbb{R}^4 reads M=adj K\mathbb{M}=\mathrm{adj}\,K, and on {det⁡M>0}\{\det\mathbb{M}>0\} it is solved identically by K~=det⁡M M−1\tilde K=\sqrt{\det\mathbb{M}}\,\mathbb{M}^{-1}. Realizability with nondegenerate second fundamental form is therefore the single equation F(g)=Cod(K~(g))=0F(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 FF as a nonlinear operator on metrics. At every nondegenerate point its principal symbol has rank two, with kernel exactly the symbol of the linearised immersions, so FF is injective modulo immersions everywhere and nowhere elliptic modulo diffeomorphisms alone. The nine Codazzi components carry four identities, leaving five conditions at every nondegenerate point, and Cartan's test holds with equality, in exact arithmetic, off a proper real-algebraic set. On a closed manifold the zero set is nowhere dense in the nondegenerate domain. At the round S3S^3 the linearisation on transverse-traceless tensors is Cod∘12(−Δ+2)\mathrm{Cod}\circ\frac{1}{2}(-\Delta+2), with singular values n(n2−1)/2n(n^2-1)/\sqrt{2} and ∥DF h∥2≥288∥h∥2\|DF\,h\|^2\ge 288\|h\|^2. For metrics foliated by round 2-spheres we solve the first-order problem in closed form and write down all hypersurfaces swept by round 2-spheres; the ℓ=1\ell=1 direction is realized exactly by tilting the spheres' 3-planes. We treat the degenerate locus by two polynomial forms, and give Gauss–Codazzi as the flatness of two su(2)\mathfrak{su}(2) connections: on the nondegenerate domain, realizability is equivalent to a single parallel spinor. Every displayed result is checked by a deposited script.

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Version history

  1. v1Submitted by Jay ThrussellInitial depositSupersededOct 29, 2026
  2. v2Submitted by Jay ThrussellSubmitted as a new versionCurrentOct 30, 2026