Holonomy and parallel spinors in Lorentzian geometry

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The holonomy group, a crucial concept in studying geometric structures on smooth manifolds, describes parallel objects in tensor bundles and other geometric vector bundles, including the spinor bundle. A classical question in differential geometry is which groups can occur as the holonomy group of a semi-Riemannian manifold and which permit parallel spinors. This question is essentially resolved for Riemannian manifolds, where the holonomy group is completely reducible. This dissertation addresses the situation for Lorentzian manifolds, which are significant in modern physics and may have holonomy representations that are not completely reducible. Some manifolds exhibit indecomposable yet non-irreducible holonomy representations, which are vital for the existence of parallel spinors. The first chapter introduces this topic and proves a decomposition theorem for parallel spinors. The second chapter details Lorentzian manifolds with indecomposable, non-irreducible holonomy representations, noting that their holonomy group is part of the parabolic group, with the screen holonomy being essential. A construction method for manifolds where the screen holonomy is a Riemannian holonomy group is also presented. The third chapter explores the existence of parallel spinors and their implications for the holonomy group. Finally, the last chapter provides a partial classification of indecomposable, non-irreducible Lorentzian holonomy group