Séminaire de Géométrie

Parallel spinors on Riemannian and Lorentzian manifolds

by Bernd Ammann (Université de Regensbrug)

1180 (Bât E2) (Tours)

1180 (Bât E2)



The talk describes results in joint articles with Klaus Kröncke, Olaf Müller, Hartmut Weiss, and Frederik Witt.
We say that a Riemannian metric on \(M\) is structured if its pullback to the universal cover admits a parallel spinor. All such metrics are Ricci-flat. The holonomy of these metrics is special as these manifolds carry some additional structure, e.g. a Calabi-Yau structure or a  \(G_2\)-structure. All known compact Ricci-flat manifolds are structured.

The set of structured Ricci-flat metrics on compact manifolds is now well-understood, and we will explain this in the first part of the talk.

The set of structured Ricci-flat metrics is an open and closed subset in the space of all Ricci-flat metrics. The holonomy group is constant along connected components. The dimension of the space of parallel spinors as well. The structured Ricci-flat metrics form a smooth Banach submanifold in the space of all metrics. Furthermore the associated premoduli space is a finite-dimensional smooth manifold, and the parallel spinors form a natural bundle with metric and connection over this premoduli space.

Lorentzian manifolds with a parallel spinor are not necessarily Ricci-flat, however the rank of the Ricci tensor is at most \(1\), the image of the Ricci-endomorphism is lightlike. Helga Baum, Thomas Leistner and Andree Lischewski showed the well-posedness for an associated  Cauchy problem. Here well-posedness means that a (local) solutions exist if and only if the initial conditions satisfy some constraint equations.

We are now able to prove a conjecture by Leistner and Lischewski which states that solutions of the constraint equations on an \(n\)-dimensional Cauchy hypersurface can be obtained from curves in the moduli space of structured Ricci-flat metrics on an \((n-1)\)-dimensional closed manifold.