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

## Abstract For a Riemannian three-manifold $(\Sigma, g)$ with negative Einstein endomorphism $\mathbb M = -G$, the Gauss equation of a hypersurface of $E^4$ reads $\mathbb M = \mathrm{adj}\,K$. 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 one equation, $F(g) = \mathrm{Cod}(\tilde K(g)) = 0$. This reduction is classical (Beez, Killing, Thomas), and the zero set of $F$ has been characterized intrinsically (Rivertz; Agaoka and Hashinaga). We study $F$ as a nonlinear operator on metrics. - **The symbol.** At every nondegenerate point the principal symbol of $F$ has rank two. Its kernel is the diffeomorphism directions plus the normal direction $K$: exactly the symbol of the linearised immersions, at every covector and on both branches. So $F$ is injective modulo the linearised immersions everywhere, and is nowhere elliptic modulo diffeomorphisms alone. - **Transverse-traceless tensors.** The symbol is injective on transverse-traceless tensors at $\xi$ iff $\xi^{\mathsf T}(\sigma_1g - K)\xi \neq 0$. So the transverse-traceless slice serves exactly where the first Newton tensor is definite, and a slice adapted to $K$ serves everywhere. - **The count.** The nine Codazzi components satisfy four identities, which leaves five conditions per point, at every nondegenerate shape operator. Cartan's test holds with equality at order three, in exact arithmetic, at sixteen shape operators, the umbilic one included, and so off a proper real-algebraic set. There the formal solutions of the symbol are exactly the linearised immersions. - **Meagreness.** On a closed three-manifold the zero set of $F$, which contains the realizable metrics, is nowhere dense in the nondegenerate domain. - **The round sphere.** On transverse-traceless tensors the linearisation is $\mathrm{Cod}\circ\tfrac12(-\Delta + 2)$, with singular values $n(n^2-1)/\sqrt2$ at level $n$. These are Branson's multipliers, and they give the bound $\lVert DF\,h\rVert^2 \geq 288\lVert h\rVert^2$. The straight path along a kernel direction leaves the realizable set at second order, while the radial lift stays in it. - **Shells.** For metrics foliated by round 2-spheres we solve in closed form the first-order problem for deformations of the radial coefficient by a shell harmonic. We write down the metrics of all hypersurfaces swept by round 2-spheres: the radial coefficient is $H^2 + (\alpha + \Phi\,b\cdot\hat n)^2$, with $b$ a free tilt of the spheres' 3-planes. So the $\ell = 1$ first-order direction is realized exactly by a tilt, with no in-plane drift of the centres. An in-plane drift realizes it together with a shear of the shell labels. - **The rest.** We treat the degenerate locus through two polynomial forms of the equation, one of them of the bidegree of Rivertz's equalities. We give Gauss–Codazzi as the flatness of two real $\mathfrak{su}(2)$ connections, the connection form of Lawn and Roth's two-spinor characterization; on the nondegenerate domain, realizability is equivalent to a single parallel spinor. Every displayed result is reproduced by a script; Appendix V indexes them.