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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.

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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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