PREFACE ix
In particular, we thoroughly discuss the geometries corresponding to |1|–gradings
(which can be described as classical first order G–structures) and the parabolic
contact geometries, which have an underlying contact structure. In Sections 4.4
and 4.5, we discuss two general constructions relating geometries of different types,
the construction of correspondence spaces and twistor spaces, and analogs of the
Fefferman construction.
The developments in Chapter 5 admit two interpretations. On the one hand,
via the notion of Weyl structures, we associate to any parabolic geometry a class
of distinguished connections and we define classes of distinguished curves. On the
other hand, the data associated to a Weyl structure offer an equivalent description
of the canonical Cartan connection in terms of objects associated to the underlying
structure. In this way, one also obtains a more explicit description of the canon-
ical Cartan connections. Throughout Sections 5.2 and 5.3 we also discuss various
applications of the theory developed in the book.
The first part of the book (Chapters 1 and 2) provides necessary background
and motivation. Chapter 1 is general and rather elementary and should be digestible
and enjoyable even for newcomers. Here Cartan’s concept of “curved analogs” of
Klein’s homogeneous spaces and also the related general calculi are explained us-
ing the effective general language of Lie groups and Lie algebras but no structure
theory. As mentioned before, some of the material presented in this discussion is
not easy to find in the literature. Section 1.6 contains an explicit and elementary
treatment of conformal (pseudo)–Riemannian structures. Apart from motivating
further developments, this also indicates clearly that a deep understanding of the
algebraic structure of the algebras and groups of symmetries in question is the key
to further progress. This naturally leads to Chapter 2, which contains background
material on semisimple Lie algebras and Lie groups. While the material we cover
in this chapter is certainly available in book form in many places, there are some
unusual aspects. The main point is that, apart from the complex theory, we also
discuss the structure theory and representation theory of real semisimple Lie alge-
bras. The real theory is typically scattered in the textbooks among the advanced
topics and hence rather difficult to learn quickly elsewhere. In this way, the first
part of the book makes the whole project more or less self–contained. In addi-
tion, it should be of separate interest as well. As an important counterpart to the
theory developed in Chapter 2 we provide tables containing the central structural
information on semisimple Lie algebras in Appendix B.
The second volume will be devoted to invariant differential operators for par-
abolic geometries, in particular, the technique of BGG–sequences, and several ap-
plications. While the links of the Cartan geometry to the more easily visible and
understandable underlying structures are among the main targets of the first vol-
ume, the second one will treat the Cartan connections as given abstract data. This
will further underline the algebraic and cohomological character of the available
tools and methods.
Suggestions for reading. We have tried to design the book in a way which
allows fruitful reading for people with different interests. Readers interested in one
or a few specific examples of the geometries covered by the general theory could
start reading the parts of the fourth and fifth chapters devoted to the structures in
question, and return to the earlier chapters to get background or general results and
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