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Parabolic Geometries I - Background and General Theory (Hardcover)
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Parabolic Geometries I - Background and General Theory (Hardcover)
Series: Mathematical Surveys and Monographs
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Parabolic geometries encompass a very diverse class of geometric
structures, including such important examples as conformal,
projective, and almost quaternionic structures, hypersurface type
CR-structures and various types of generic distributions. The
characteristic feature of parabolic geometries is an equivalent
description by a Cartan geometry modeled on a generalized flag
manifold (the quotient of a semisimple Lie group by a parabolic
subgroup). Background on differential geometry, with a view towards
Cartan connections, and on semisimple Lie algebras and their
representations, which play a crucial role in the theory, is
collected in two introductory chapters. The main part discusses the
equivalence between Cartan connections and underlying structures,
including a complete proof of Kostant's version of the Bott - Borel
- Weil theorem, which is used as an important tool. For many
examples, the complete description of the geometry and its basic
invariants is worked out in detail. The constructions of
correspondence spaces and twistor spaces and analogs of the
Fefferman construction are presented both in general and in several
examples. The last chapter studies Weyl structures, which provide
classes of distinguished connections as well as an equivalent
description of the Cartan connection in terms of data associated to
the underlying geometry. Several applications are discussed
throughout the text.
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