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Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems

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Geometric Flows and the Geometry of Space-time

Abstract

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 describe the algebraic classification of connected Lorentzian holonomy groups. In particular, we specify the holonomy groups of locally indecomposable Lorentzian spin manifolds with a parallel spinor field. In the second part we explain new methods for the construction of globally hyperbolic Lorentzian manifolds with special holonomy based on the solution of certain Cauchy problems for PDEs that are imposed by the existence of a parallel lightlike vector field or a parallel lightlike spinor field with initial conditions on a spacelike hypersurface. Thereby, we derive a second order evolution equation of Cauchy-Kowalevski type that can be solved in the analytic setting as well as an appropriate first order quasilinear hyperbolic system that yields a solution in the smooth case.

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Notes

  1. 1.

    We assume all manifolds to be smooth, connected and without boundary.

  2. 2.

    In fact, the holonomy algebra of every torsionfree connection is a Berger algebra.

  3. 3.

    We call an orthonormal basis (s 1, …, s n) of (T x M, g x) adapted, if g x(s j, s j) = −1 for 1 ≤ j ≤ p and g x(s j, s j) = 1 for p + 1 ≤ j ≤ n.

  4. 4.

    If n = p + q is odd, the two possible restrictions are equivalent as Spin(p, q)-representations.

  5. 5.

    This is in fact true in any signature.

References

  1. D.V. Alekseevsky, B.N. Kimmelfeld, Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funct. Anal. Appl. 9(2), 5–11 (1975)

    MathSciNet  Google Scholar 

  2. D. Alekseevsky, V. Cortés, A. Galaev, T. Leistner, Cones over pseudo-Riemannian manifolds and their holonomy. J. Reine Angew. Math. 635, 23–69 (2009)

    MathSciNet  MATH  Google Scholar 

  3. B. Ammann, A. Moroianu, S. Moroianu, The Cauchy problems for Einstein metrics and parallel spinors. Commun. Math. Phys. 320(1), 173–198 (2013)

    Article  MathSciNet  Google Scholar 

  4. C. Bär, Real Killing spinors and holonomy. Commun. Math. Phys. 154(3), 509–521 (1993)

    Article  MathSciNet  Google Scholar 

  5. C. Bär, P. Gauduchon, A. Moroianu, Generalized cylinders in semi-Riemannian and Spin geometry. Math. Z. 249(3), 545–580 (2005)

    Article  MathSciNet  Google Scholar 

  6. C. Bär, N. Ginoux, F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics (European Mathematical Society, Zürich, 2007)

    Google Scholar 

  7. R. Bartnik, J. Isenberg, The constraint equations, in The Einstein Equations and the Large Scale Behavior of Gravitational Fields (Birkhäuser, Basel, 2004), pp. 1–38

    MATH  Google Scholar 

  8. H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten. Teubner-Texte zur Mathematik, vol. 41 (Teubner-Verlagsgesellschaft, Leipzig, 1981)

    Google Scholar 

  9. H. Baum, Complete Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom. 7(3), 205–226 (1989)

    Article  MathSciNet  Google Scholar 

  10. H. Baum, Odd-dimensional Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom. 7(2), 141–153 (1989)

    Article  MathSciNet  Google Scholar 

  11. H. Baum, A remark on the spectrum of the Dirac operator on pseudo-Riemannian spin maniofolds. Technical Report 136, SFB 288-Preprint, 1994

    Google Scholar 

  12. H. Baum, Eichfeldtheorie. Eine Einführung in die Differentialgeometrie auf Faserbündeln, 2nd revised edition (Springer Spektrum, Heidelberg, 2014)

    Google Scholar 

  13. H. Baum, O. Müller, Codazzi spinors and globally hyperbolic manifolds with special holonomy. Math. Z. 258(1), 185–211 (2008)

    Article  MathSciNet  Google Scholar 

  14. H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistors and Killing spinors on Riemannian manifolds. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 124 (B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991)

    Google Scholar 

  15. H. Baum, K. Lärz, T. Leistner, On the full holonomy group of Lorentzian manifolds. Math. Z. 277(3–4), 797–828 (2014)

    Article  MathSciNet  Google Scholar 

  16. H. Baum, T. Leistner, A. Lischewski, Cauchy problems for Lorentzian manifolds with special holonomy. Differ. Geom. Appl. 45, 43–66 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ya. V. Bazaikin, Globally hyperbolic Lorentzian spaces with special holonomy groups. Siberian Math. J. 50(4), 567579 (2009)

    Article  MathSciNet  Google Scholar 

  18. L. Bérard-Bergery, A. Ikemakhen, On the holonomy of Lorentzian manifolds, in Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990). Proc. Sympos. Pure Math., vol. 54 (American Mathematical Society, Providence, RI, 1993), pp. 27–40

    Google Scholar 

  19. M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)

    Article  MathSciNet  Google Scholar 

  20. M. Berger, Les espaces symétriques noncompacts. Ann. Sci. École Norm. Sup. (3) 74, 85–177 (1957)

    Article  MathSciNet  Google Scholar 

  21. A.N. Bernal, M. Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257(1), 43–50 (2005)

    Article  MathSciNet  Google Scholar 

  22. A.L. Besse, Einstein Manifolds (Springer, Berlin/Heidelberg/New York, 1987)

    Book  Google Scholar 

  23. N. Bezvitnaya, Lightlike foliations on Lorentzian manifolds with weakly irreducible holonomy algebra (2005). Preprint. arXiv:math.DG/0506101

    Google Scholar 

  24. R.L. Bryant, Metrics with exceptional holonomy. Ann. Math. (2) 126(3), 525–576 (1987)

    Article  MathSciNet  Google Scholar 

  25. R.L. Bryant, Classical, exceptional, and exotic holonomies: a status report. Séminaires Conrès Soc. Math. France 1, 93–165 (1996)

    MathSciNet  MATH  Google Scholar 

  26. R.L. Bryant, Recent advances in the theory of holonomy. Séminaire Bourbaki, Asterisque 51, 525–576 (1999)

    Google Scholar 

  27. R.L. Bryant, Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999). Sémin. Congr., vol. 4 (Société Mathématique de France, Paris, 2000), pp. 53–94

    Google Scholar 

  28. R.L. Bryant, Some remarks on G 2-structures, in Proceedings of Gökova Geometry-Topology Conference 2005 (Gökova Geometry/Topology Conference (GGT), Gökova, 2006), pp. 75–109

    Google Scholar 

  29. M. Cahen, N. Wallach, Lorentzian symmetric spaces. Bull. Am. Math. Soc. 79, 585–591 (1970)

    Article  MathSciNet  Google Scholar 

  30. A.M. Candela, J.L. Flores, M. Sanchez, On general plane fronted waves. Gen. Relat. Gravit. 35(4), 631–649 (2003)

    Article  MathSciNet  Google Scholar 

  31. E. Cartan, La Geometrie des Espaces de Riemann. Memorial des Sciences Mathematiques, Fasc. IX, Ch. VII, Sec. II (1925)

    Google Scholar 

  32. E. Cartan, Sur une classe remarquable d’espaces Riemann. Bull. Soc. Math. France 54, 214–264 (1926)

    Article  MathSciNet  Google Scholar 

  33. E. Cartan, Les groups d’holonomie des espaces généralisés. Acta Math. 48, 1–42 (1926)

    Article  Google Scholar 

  34. G. Darmois, Les equations de la gravitation Einsteinienne. Memorial des Sciences Mathématiques, vol. XXV (Gauthier Villars, Paris, 1927)

    Google Scholar 

  35. G. de Rham, Sur la reductibilité d’un espace de Riemann. Comment. Math. Helv. 26, 328–344 (1952)

    Article  MathSciNet  Google Scholar 

  36. A.J. Di Scala, C. Olmos, The geometry of homogeneous submanifolds of hyperbolic space. Math. Z. 237(1), 199–209 (2001)

    Article  MathSciNet  Google Scholar 

  37. J.L. Flores, M. Sanchez, Causality and conjugated points in general plane waves. Class. Quant. Gravity 20, 2275–2291 (2003)

    Article  Google Scholar 

  38. G.B. Folland, Introduction to Partial Differential Equations, 2nd edn. (Princeton University Press, Princeton, NJ, 1995)

    MATH  Google Scholar 

  39. Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)

    Article  MathSciNet  Google Scholar 

  40. F.G. Friedlander, The Wave Equation on Curved Space Time (Cambridge University Press, Cambridge, MA, 1975)

    MATH  Google Scholar 

  41. T. Friedrich, Dirac Operators in Riemannian Geometry. Transl. from the German by Andreas Nestke (American Mathematical Society, Providence, RI, 2000)

    Google Scholar 

  42. H. Friedrich, A. Rendall, The Cauchy problem for the Einstein equations, in Einstein’s Field Equations and Their Physical Implications. Lecture Notes in Physics, vol. 540 (Springer, Berlin, 2000), pp. 127–223

    Chapter  Google Scholar 

  43. A.S. Galaev, Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups. Rend. Circ. Mat. Palermo (2) Suppl. 79, 87–97 (2006)

    Google Scholar 

  44. A.S. Galaev, Metrics that realize all Lorentzian holonomy algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1025–1045 (2006)

    Article  MathSciNet  Google Scholar 

  45. A.S. Galaev, Holonomy groups of Lorentzian manifolds. Russ. Math. Surv. 70(2), 249–298 (2015)

    Article  MathSciNet  Google Scholar 

  46. A.S. Galaev, T. Leistner, Holonomy groups of Lorentzian manifolds: classification, examples, and applications, in Recent Developments in Pseudo-Riemannian Geometry. ESI Lectures in Mathematics and Physics, pp. 53–96 (European Mathematical Society, Zürich, 2008)

    Google Scholar 

  47. S. Gallot, Équations différentielles caractéristiques de la sphère. Ann. Sci. École Norm. Sup. (4) 12(2), 235–267 (1979)

    Article  MathSciNet  Google Scholar 

  48. R. Geroch, Domain of dependence. J. Math. Phys. 11, 437–449 (1970)

    Article  MathSciNet  Google Scholar 

  49. G.S. Hall, D.P. Lonie, Holonomy groups and spacetimes. Classical Quantum Gravity 17(6), 1369–1382 (2000)

    Article  MathSciNet  Google Scholar 

  50. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34 (American Mathematical Society, , Providence, RI, 2001)

    Google Scholar 

  51. D.D. Joyce, Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs (Oxford Univerity Press, Oxford, 2000)

    MATH  Google Scholar 

  52. D.D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, vol. 12 (Oxford University Press, Oxford, 2007)

    Google Scholar 

  53. S. Karigiannis, Deformations of G 2 and Spin(7) structures. Can. J. Math. 57(5), 1012–1055 (2005)

    Article  MathSciNet  Google Scholar 

  54. S. Karigiannis, Flows of G 2-structures. I. Q. J. Math. 60(4), 487–522 (2009)

    Article  MathSciNet  Google Scholar 

  55. M. Karoubi, Algèbres de Clifford et K-théorie. Ann. Sci. École Norm. Sup. (4) 1, 161–270 (1968)

    Article  MathSciNet  Google Scholar 

  56. N. Koiso, Hypersurfaces of Einstein manifolds. Ann. Sci. École Norm. Sup. (4) 14(4), 433–443 (1981)

    Article  MathSciNet  Google Scholar 

  57. K. Lärz, Global Aspects of Holonomy in Pseudo-Riemannian Geometry. PhD thesis, Humboldt-Universität zu Berlin, 2011. Available at http://edoc.hu-berlin.de/dissertationen/

  58. H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, Princeton, 1989)

    MATH  Google Scholar 

  59. T. Leistner, Berger algebras, weak-Berger algebras and Lorentzian holonomy, 2002. SFB 288-Preprint no. 567, ftp://ftp-sfb288.math.tu-berlin.de/pub/Preprints/preprint567.ps.gz

  60. T. Leistner, Holonomy and parallel spinors in Lorentzian geometry. PhD thesis, Berlin: Logos-Verlag; Berlin: Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultät II (Dissertation). xii, 178 p. , 2003

    Google Scholar 

  61. T. Leistner, On the classification of Lorentzian holonomy groups. J. Differ. Geom. 76(3), 423–484 (2007)

    Article  MathSciNet  Google Scholar 

  62. T. Leistner, A. Lischewski, Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors. J. Geom. Anal. (2017). https://doi.org/10.1007/s12220-017-9941-x

  63. T. Leistner, D. Schliebner, Completeness of compact Lorentzian manifolds with abelian holonomy. Math. Ann. 364(3–4), 1469–1503 (2016)

    Article  MathSciNet  Google Scholar 

  64. A. Lichnerowicz, Problèmes globaux en mécanique relativiste. Actual. Sci. Ind. 833. Exposés de géométrie. XII. Hermann et Cie, Paris, 1939

    Google Scholar 

  65. A. Lischewski, The Cauchy problem for parallel spinors as first-order symmetric hyperbolic system (March 2015). Preprint. arXiv:1503.04946

    Google Scholar 

  66. E. Minguzzi, M. Sánchez, The causal hierarchy of spacetimes, in Recent Developments in Pseudo-Riemannian Geometry. ESI Lectures in Mathematics and Physics (European Mathematical Society, Zürich, 2008), pp. 299–358

    Google Scholar 

  67. A. Moroianu, U. Semmelmann, Parallel spinors and holonomy groups. J. Math. Phys. 41(4), 2395–2402 (2000)

    Article  MathSciNet  Google Scholar 

  68. B. O’Neill, Semi-Riemannian Geometry (Academic, Cambridge, 1983)

    MATH  Google Scholar 

  69. H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, in Global Differential Geometry and Global Analysis (Berlin, 1990). Lecture Notes in Mathematics, vol. 1481 (Springer, Berlin, 1991), pp. 192–198

    Chapter  Google Scholar 

  70. H. Ringström, The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics (European Mathematical Society, Zürich, 2009)

    Google Scholar 

  71. S.M. Salamon, Riemannian Geometry and Holonomy Groups. Pitmann Research Lecture Notes, vol. 201 (Longman Scientific & Technical, Harlow, 1989)

    Google Scholar 

  72. J.F. Schell, Classification of 4-dimensional Riemannian spaces. J. Math. Phys. 2, 202–206 (1960)

    Article  MathSciNet  Google Scholar 

  73. R. Shaw, The subgroup structure of the homogeneous Lorentz group. Quart. J. Math. Oxford 21, 101–124 (1970)

    Article  MathSciNet  Google Scholar 

  74. M.E. Taylor, Partial Differential Equations III. Nonlinear Equations. Applied Mathematical Sciences, vol. 117, 2nd edn. (Springer, New York, 2011)

    Google Scholar 

  75. M.Y. Wang, Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7(1), 59–68 (1989)

    Article  MathSciNet  Google Scholar 

  76. M.Y. Wang, On non-simply connected manifolds with non-trivial parallel spinors. Ann Global Anal. Geom. 13, 31–42 (1995)

    Article  MathSciNet  Google Scholar 

  77. H. Wu, On the de Rham decomposition theorem. Illinois J. Math. 8, 291–311 (1964)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Humboldt University Berlin, Department of Mathematics, Berlin, Germany

    Helga Baum

  2. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, Australia

    Thomas Leistner

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  1. Helga Baum
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  2. Thomas Leistner
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Correspondence to Helga Baum or Thomas Leistner .

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Editors and Affiliations

  1. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Vicente Cortés

  2. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Klaus Kröncke

  3. Department of Physics, University of Hamburg, Hamburg, Germany

    Jan Louis

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Baum, H., Leistner, T. (2018). Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems. In: Cortés, V., Kröncke, K., Louis, J. (eds) Geometric Flows and the Geometry of Space-time. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01126-0_1

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