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Holonomy and parallel spinors in Lorentzian geometry. (English) Zbl 1088.53032
Berlin: Logos-Verlag; Berlin: Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultät II (Dissertation) (ISBN 3-8325-0472-9/pbk). xii, 178 p. (2003).
The author studies Lorentzian manifolds and their holonomy groups under the condition that they have parallel spinors. Firstly, he shows that a semi-Riemannian product manifold admits a parallel spinor if and only if every factor admits a parallel spinor. Using the Wu-decomposition theorem and the vector field associated to the parallel spinor, he solves the existence problem of parallel spinors for Lorentzian manifolds which decompose into irreducible factors. For indecomposable, non-irreducible Lorentzian manifolds, their holonomy group is contained in the parabolic group \(({\mathbb R}^* \times SO(n)) \times {\mathbb R}^n\); and it is proved that the \(\text{SO}(n)\)-component of the holonomy group is the holonomy group of a certain metrical vector bundle which is called the screen bundle. The existence of parallel spinors implies that the \({\mathbb R}^*\)-component of the holonomy has to be trivial and the \(SO(n)\)-component cannot be Abelian. If the latter contains \(\text{SU}(n/2)\), then the existence of a parallel spinor implies that it is equal to \(\text{SU}(n/2)\).
The author studies the following question: Which Lie groups can occur as \(\text{SO}(n)\)-component of the holonomy group of a simply connected indecomposable, non-irreducible Lorentzian manifold? He obtains an algebraic criterion, based on the first Bianchi identity, which is analogous to the Berger criterion, but restricted to the \(\text{so}(n)\)-component; and he proves that if \(G\) is the \(\text{SO}(n)\)-component of an indecomposable, non-irreducible Lorentzian manifold, then it is a Riemannian holonomy group, provided that it acts irreducibly and is simple, or provided that \(G\subset U(n/2)\). Therefore, an indecomposable Lorentzian manifold with parallel spinor vector field does not have a holonomy group of coupled type.

53C29 Issues of holonomy in differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C27 Spin and Spin\({}^c\) geometry