Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on $$M\times (a,b)$$ M × ( a , b ) (resp. $$M\times S^1$$ M × S 1 ).

Stability of Riemannian manifolds with Killing spinors

Riemannian manifolds with nonzero Killing spinors are Einstein manifolds. Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in [Stable and unstable Einstein warped products, preprint (2015), arXiv:1507.01782v1 ]. In this paper, we obtain a new proof for this stability result by using a Bochner-type formula in [X. Dai, X. Wang and G. Wei, On the stability of Riemannian manifold with parallel spinors, Invent. Math. 161(1) (2005) 151–176; M. Wang, Preserving parallel spinors under metric deformations, Indiana Univ. Math. J. 40 (1991) 815–844]. Moreover, existence of real Killing spinors is closely related to the Sasaki–Einstein structure. A regular Sasaki–Einstein manifold is essentially the total space of a certain principal [Formula: see text]-bundle over a Kähler–Einstein manifold. We prove that if the base space is a product of two Kähler–Einstein manifolds then the regular Sasaki–Einstein manifold is unstable. This provides us many new examples of unstable manifolds with real Killing spinors.

Generalized Killing spinors on Einstein manifolds

We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence of compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi–Yau, hyperkähler, G2 and Spin(7) manifolds.

THE ENERGY-MOMENTUM TENSOR ON Spinc MANIFOLDS

On Spinc manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spinc manifold. Using the notion of generalized cylinders, we derive the variational formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spinc Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.