# Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors

## TL;DR

This paper establishes well-posedness for the Cauchy problem of parallel null vector fields in Lorentzian manifolds, classifies Riemannian manifolds with specific holonomy, and links these to generalized Killing spinors and special holonomy flow equations.

## Contribution

It introduces a hyperbolic evolution framework for Lorentzian geometry problems and classifies Riemannian manifolds with certain holonomy and spinor properties.

## Key findings

- Well-posedness of the Cauchy problem for null vector fields
- Classification of Riemannian manifolds with special holonomy
- New local forms for Lorentzian metrics with parallel null spinors

## Abstract

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups these flow equations have been investigated in the literature before in other contexts. As an application, the results provide a classification of Riemannian manifolds admitting imaginary generalised Killing spinors. We will also give new local normal forms for Lorentzian metrics with parallel null spinor in any dimension.