Cauchy Spinors on 3-Manifolds

Abstract

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 hyperkähler 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.