**Mathematical Surveys and Monographs**

Volume: 154;
2009;
628 pp;
Hardcover

MSC: Primary 53; 58;

Print ISBN: 978-0-8218-2681-2

Product Code: SURV/154

List Price: $120.00

Individual Member Price: $96.00

**Electronic ISBN: 978-1-4704-1381-1
Product Code: SURV/154.E**

List Price: $120.00

Individual Member Price: $96.00

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#### Supplemental Materials

# Parabolic Geometries I: Background and General Theory

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*Andreas Čap; Jan Slovák*

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.

#### Table of Contents

# Table of Contents

## Parabolic Geometries I: Background and General Theory

- Contents v7 free
- Preface vii9 free
- Part 1. Background 113 free
- Chapter 1. Cartan geometries 315
- Chapter 2. Semisimple Lie algebras and Lie groups 141153
- Part 2. General Theory 231243
- Chapter 3. Parabolic geometries 233245
- Chapter 4. A panorama of examples 363375
- Chapter 5. Distinguished connections and curves 497509
- Appendix A. Other prolongation problems 599611
- Appendix B. Tables 607619
- Bibliography 617629
- Index 623635 free

#### Readership

Graduate students and research mathematicians interested in parabolic geometry, conformal geometry, almost quaternionic structures, and CR-structures.

#### Reviews

An excellent book. Serving both as a timely introduction to parabolic geometry and as a general introductory work for Lie groups and Cartan geometries. ... This review cannot do justice to the power and generality of parabolic geometry theory, but this book certainly does.

-- Mathematical Reviews