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Holonomy and Parallel Spinors in Lorentzian Geometry (Paperback)
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Holonomy and Parallel Spinors in Lorentzian Geometry (Paperback)
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The group of parallel displacements, the so-called holonomy group,
is an important tool in order tostudy geometric structures on a
smooth manifold. It describes parallel objects in tensor bundles as
well as in other geometric vector bundles, such as the spinor
bundle. The question: Which groups may occur as holonomy group of a
semi-Riemannian manifold and which of these admit parallel spinors?
is a classical one in differential geometry. It is essentially
solved for Riemannian manifolds, because here the holonomy group
acts completely reducible. This dissertation is an aproach to
answer this question for Lorentzian manifolds.Lorentzian
manifoldsare relevant for modern physics and their holonomy
representation is not necessarily completely reducible. There are
manifolds with indecomposable, but non-irreducible holonomy
representation. These are interesting for the existence of parallel
spinors. This is explained in the mainly introductory first
chapter. Here also a decomposition theorem for the existence of
parallel spinors is proven. Lorentzian manifolds with
indecomposable, but non-irreducible holonomy representation are
described in detail in the second chapter. Their holonomy group is
contained in the parabolic group and its essential part is the
screen holonomy. A construction method is given for manifolds, for
which the screen holonomy is a Riemannian holonomy group. In the
third chapter the existence of parallel spinors and its
consequences for the holonomy group and the screen holonomy are
studied. In the last chapter a partial classification of
indecomposable, non-irreducible Lorentzian holonomy groups is given
by proving that in most of the cases the screen holonomy has to be
a Riemannian holonomy. The consequences for the existence of
parallel spinors are drawn. This classification result uses methods
of representation theory of real, semisimple Lie algebras. The main
facts of this theory are briefly explained in the appendix.
General
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