Cauchy problems for Lorentzian manifolds with special holonomy

TL;DR

This paper explores the geometric conditions under which Lorentzian manifolds with special holonomy, including parallel null vector fields and spinors, can be extended from Riemannian manifolds satisfying specific constraints.

Contribution

It establishes a correspondence between Riemannian manifolds satisfying certain constraints and Lorentzian manifolds with special holonomy, including null vector fields and spinors, and provides explicit examples.

Findings

Constraint conditions on Riemannian hypersurfaces derived

Extension of real analytic Riemannian manifolds to Lorentzian manifolds with special holonomy proven

Examples of geodesically complete manifolds satisfying constraints provided

Abstract

On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector field. Similarly, every parallel null spinor on a Lorentzian manifold induces an imaginary generalised Killing spinor on a space-like hypersurface. Then, based on the fact that a parallel spinor field induces a parallel vector field, we can apply the first result to prove: every real analytic Riemannian manifold carrying a real analytic, imaginary generalised Killing spinor can be extended to a Lorentzian manifold with a parallel null spinor. Finally, we give examples of geodesically complete Riemannian manifolds satisfying the constraint conditions.