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.

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