Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor

TL;DR

This paper explores the geometry and normal forms of pseudo-Riemannian metrics with parallel spinor fields, focusing on cases where the Ricci tensor vanishes, highlighting their unique geometric properties.

Contribution

It provides a detailed analysis of the geometry and normal forms of such metrics, especially in relation to Ricci-flatness and parallel spinor fields in specific dimensions.

Findings

Characterization of pseudo-Riemannian metrics with parallel spinors

Normal forms for these metrics in certain dimensions

Conditions under which Ricci tensor vanishes in this context

Abstract

I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci tensor vanish (which, for pseudo-Riemannian manifolds, is not an automatic consequence of the existence of a nontrivial parallel spinor field).