math.DG — Differential Geometry

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

Contributed by Jay Thrussell

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 on metrics. At every nondegenerate point its principal symbol has rank two, with kernel exactly the symbol of the linearised immersions, so $F$ 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 $S^3$ the linearisation on transverse-traceless tensors is $\mathrm{Cod}\circ\frac{1}{2}(-\Delta+2)$, with singular values $n(n^2-1)/\sqrt{2}$ and $\|DF\,h\|^2\ge 288\|h\|^2$. 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 $\ell=1$ 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 $\mathfrak{su}(2)$ connections: on the nondegenerate domain, realizability is equivalent to a single parallel spinor. Every displayed result is checked by a deposited script.