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Parabolic Geometries I: Background and General Theory
 
Andreas Čap Universität Wien, Wien, Austria and International Erwin Schrödinger Institute for Mathematical Physics, Wien, Austria
Jan Slovák Masaryk University, Brno, Czech Republic
Parabolic Geometries I
Hardcover ISBN:  978-0-8218-2681-2
Product Code:  SURV/154
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1381-1
Product Code:  SURV/154.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-2681-2
eBook: ISBN:  978-1-4704-1381-1
Product Code:  SURV/154.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
Parabolic Geometries I
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Parabolic Geometries I: Background and General Theory
Andreas Čap Universität Wien, Wien, Austria and International Erwin Schrödinger Institute for Mathematical Physics, Wien, Austria
Jan Slovák Masaryk University, Brno, Czech Republic
Hardcover ISBN:  978-0-8218-2681-2
Product Code:  SURV/154
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-1381-1
Product Code:  SURV/154.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-0-8218-2681-2
eBook ISBN:  978-1-4704-1381-1
Product Code:  SURV/154.B
List Price: $254.00 $191.50
MAA Member Price: $228.60 $172.35
AMS Member Price: $203.20 $153.20
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1542009; 628 pp
    MSC: 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 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.

    Readership

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

  • 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
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1542009; 628 pp
MSC: 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 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.

Readership

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

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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