Non-locally homogeneous three-manifolds carrying affine Killing spinors

Determine whether there exist oriented Riemannian 3-manifolds admitting an affine Killing spinor—namely, a nonzero spinor satisfying \(\nabla_X\psi=(\lambda\,Ric+\mu\,Id)(X)\cdot\psi\) for constants \(\lambda\neq0\) and \(\mu\in\mathbb{R}\)—that are not locally homogeneous.

Background

The paper studies affine Killing spinors (AKS), whose associated endomorphism is a constant linear combination of the Ricci endomorphism and the identity. It proves that, in dimension three, every curvature-homogeneous Riemannian manifold carrying an AKS is locally homogeneous; moreover, the authors note that all known connected Riemannian spin manifolds carrying an AKS have Ricci endomorphism with at most two distinct eigenvalues, and consequently all known oriented three-dimensional examples are locally homogeneous.

The unresolved issue is whether the local homogeneity conclusion holds for every oriented Riemannian three-manifold admitting an AKS, or whether examples exist outside the currently known locally homogeneous class. Such an example would provide an AKS manifold that is not locally homogeneous and would therefore lie beyond the examples discussed in the paper.