Cauchy spinors on $3$-manifolds

Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a non-elliptic, non-linear equation in terms of a symmetric $2$-tensor on $M$. When $M$ is compact and has positive scalar curvature, the linearized equation has finite dimensional kernel. Four families of solutions are known on the round $3$-sphere $\mathbb{S}^3$. We study the linearized equation in the vicinity of these solutions and we construct as a byproduct an incomplete hyperk\"ahler metric on $\mathbb{S}^3\times \mathbb{R}$ closely related to the Euclidean Taub-NUT metric on $\mathbb{R}^4$. On $\mathbb{S}^3$ there do not exist other solutions which either are constant in a left (or right) invariant frame, have three distinct constant eigenvalues, or are invariant in the direction of a left (or right)-invariant eigenvector. We deduce from this last result an extension of Liebmann's sphere rigidity theorem.