Pseudo-Riemannian manifolds with recurrent spinor fields

TL;DR

This paper characterizes pseudo-Riemannian manifolds with recurrent spinor fields by analyzing their holonomy algebras, covering Riemannian, Lorentzian, and special neutral signature cases.

Contribution

It provides a holonomy-based classification of simply connected pseudo-Riemannian manifolds admitting recurrent spinor fields.

Findings

Characterization for Riemannian manifolds

Results for Lorentzian manifolds

Classification of neutral signature cases

Abstract

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.