Cauchy problems for Lorentzian manifolds with special holonomy

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.

26 pages

References in corpus (4)

• Initial Value Problem in General Relativity
• The Constraint Equations
• Generalized Killing spinors on spheres
• The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system

Cited by in corpus (7)

• Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity
• The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system
• Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors
• Parallel spinor flows on three-dimensional Cauchy hypersurfaces
• Parallel spinors on globally hyperbolic Lorentzian four-manifolds
• Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors
• Killing spinors and hypersurfaces