Generalised Killing Spinors and Euclidean Hypersurfaces

Abstract

We study when the number of spinors satisfying the same generalised Killing equation for a symmetric endomorphism field $W$ forces local realisation as a Euclidean hypersurface. Let $N=2^{\lfloor n/2\rfloor}$ be the rank of the complex spinor bundle. We show that in dimension $n\geq4$, if the complex dimension of the solution space exceeds $N/2$, then the metric and $W$ satisfy the Gauss--Codazzi equations. This is proved in the smooth category by combining the curvature defects into a single bivector and bounding the dimension of its kernel on spinors, using algebraic curvature symmetries to handle exceptional cases. Examples obtained from non-flat four-dimensional hyperkähler manifolds show that this threshold is sharp for every $n\geq4$. In dimension three, the Heisenberg group carries the maximal number of common solutions while failing the Euclidean Gauss equation. Further, in the real analytic category, we use existing extension theorems for Einstein metrics and parallel spinors to extend all common solutions to parallel spinors on a single local Ricci-flat ambient manifold. Using Wang's counts of parallel spinors, we obtain an independent proof of the sharp threshold for real analytic data.