Hardcover ISBN:  9780821826812 
Product Code:  SURV/154 
List Price:  $128.00 
MAA Member Price:  $115.20 
AMS Member Price:  $102.40 
Electronic ISBN:  9781470413811 
Product Code:  SURV/154.E 
List Price:  $120.00 
MAA Member Price:  $108.00 
AMS Member Price:  $96.00 

Book DetailsMathematical Surveys and MonographsVolume: 154; 2009; 628 ppMSC: Primary 53; 58;
Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CRstructures 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.ReadershipGraduate students and research mathematicians interested in parabolic geometry, conformal geometry, almost quaternionic structures, and CRstructures.

Table of Contents

Background

1. Cartan geometries

2. Semisimple Lie algebras and Lie groups

General theory

3. Parabolic geometries

4. A panorama of examples

5. Distinguished connections and curves

Appendix A. Other prolongation procedures

Appendix B. Tables


Additional Material

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


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

Background

1. Cartan geometries

2. Semisimple Lie algebras and Lie groups

General theory

3. Parabolic geometries

4. A panorama of examples

5. Distinguished connections and curves

Appendix A. Other prolongation procedures

Appendix B. Tables

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