Parabolic Geometries I
Background and General Theory
Seiten
2024
American Mathematical Society (Verlag)
978-1-4704-7822-3 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-7822-3 (ISBN)
Delving into the interplay of differential geometry and Lie algebra theory, the text unpacks parabolic geometries through Cartan connections and their underlying structures. Detailed examples illustrate conformal, projective, and CR-structures, revealing key invariants and constructions that bridge abstract theory with concrete models.
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.
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.
Andreas Cap, Universitat Wien, Austria, and International Erwin Schrodinger Institute for Mathematical Physics, Wien, Austria, and Jan Slovak, Masaryk University, Brno, Czech Republic.
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
| Erscheinungsdatum | 14.05.2025 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 1-4704-7822-6 / 1470478226 |
| ISBN-13 | 978-1-4704-7822-3 / 9781470478223 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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