# The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system

@article{Lischewski2015TheCP, title={The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system}, author={Andree Lischewski}, journal={arXiv: Differential Geometry}, year={2015} }

We prove that a smooth Riemannian manifold admitting an imaginary generalized Killing spinor whose Dirac current satisfies an additional algebraic constraint condition can be embedded as spacelike Cauchy hypersurface in a smooth Lorentzian manifold on which the given spinor extends to a null parallel spinor. This is in contrast to a corresponding Cauchy problem for real generalized Killing spinors into Riemannian manifolds. The construction is based on first order symmetric hyperbolic PDE…

## 9 Citations

Hyperbolic Evolution Equations, Lorentzian Holonomy, and Riemannian Generalised Killing Spinors

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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…

Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems

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This review is based on lectures given by the authors during the Summer School Geometric Flows and the Geometry of Space-Time at the University of Hamburg, September 19–23, 2016. In the first part we…

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The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian…

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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,…

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Let (M, g) be a timeand space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of M with induced Riemannian metric g and second fundamental form K. If (M, g) satisfies…

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Riemannian manifolds with positive scalar curvature play an important role in mathematics and general relativity. Obstruction and existence results are connected to index theory, bordism theory and…

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We consider spin manifolds with an Einstein metric, either Riemannian or indeﬁnite, for which there exists a Killing spinor. We describe the intrinsic geometry of nondegenerate hypersurfaces in terms…

Parallel spinors on globally hyperbolic Lorentzian four-manifolds

- MathematicsAnnals of Global Analysis and Geometry
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We investigate the differential geometry and topology of globally hyperbolic four-manifolds $(M,g)$ admitting a parallel real spinor $\varepsilon$. Using the theory of parabolic pairs recently…

## References

SHOWING 1-10 OF 18 REFERENCES

Generalized cylinders in semi-Riemannian and spin geometry

- Mathematics
- 2005

Abstract.We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result…

The Cauchy Problems for Einstein Metrics and Parallel Spinors

- Mathematics
- 2013

The restriction of a parallel spinor on some spin manifold $${\mathcal{Z}}$$ to a hypersurface $${M \subset \mathcal{Z}}$$ is a generalized Killing spinor on M. We show, conversely, that in the real…

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, I

- Mathematics
- 1972

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented. A number of sharp regularity and smoothness properties of the solutions are…

Wave Equations on Lorentzian Manifolds and Quantization

- Mathematics
- 2007

This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one…

Codazzi spinors and globally hyperbolic manifolds with special holonomy

- Mathematics
- 2007

In this paper, we describe the structure of Riemannian manifolds with a special kind of Codazzi spinors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy surface…

General Relativity and the Einstein Equations

- Mathematics
- 2009

FOREWORD ACKNOWLEDGEMENTS 1. Lorentzian Geometry 2. Special Relativity 3. General Relativity and the Einstein Equations 4. Schwarzschild Space-time and Black Holes 5. Cosmology 6. Local Cauchy…

The Cauchy Problem in General Relativity

- Mathematics
- 2009

After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how…

Breaking the M-waves

- Mathematics
- 1999

We present a systematic attempt at classification of supersymmetric M-theory vacua with zero flux; that is, eleven-dimensional lorentzian manifolds with vanishing Ricci curvature and admitting…

On the maximal superalgebras of supersymmetric backgrounds

- Mathematics
- 2009

In this paper we give a precise definition of the notion of a maximal superalgebra of certain types of supersymmetric supergravity backgrounds, including the Freund–Rubin backgrounds, and propose a…