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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, hyper-
surface type CR-structures and various types
of generic distributions. The characteristic
feature of parabolic geometries is an equiva-
lent 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 equiva-
lence 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 impor-
tant 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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