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Parabolic geometries I. Background and general theory. (English) Zbl 1183.53002
Mathematical Surveys and Monographs 154. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-2681-2/hbk). x, 628 p. (2009).
The present book is the first of a two-volume series on Parabolic Geometries. The forthcoming second volume will be called Parabolic Geometries II: Invariant Differential Operators and Applications. A parabolic geometry is a Cartan geometry of type $$(G, P)$$, where $$G$$ is a semisimple Lie group and $$P$$ a parabolic subgroup. Such geometries encompass a very diverse class of geometric structures, including conformal, projective structures, almost quaternionic, hypersurface type CR-structures, and various types of generic distributions. The characteristic features of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold, i.e., the quotient of a semisimple Lie group by a parabolic subgroup. The book provides an extensive description of the subject, which is developed in its second part namely Chapters 3, 4 and 5.
In Chapter 3 the authors develop the basic theory of parabolic geometries, and prove the equivalence to underlying structures in the categorical sense. This is done in the setting of $$|k|$$-gradings of semisimple Lie algebras. This chapter also contains a complete proof of Kostant’s version of the Bott-Borel-Weil theorem which is used as an important tool.
In Chapter 4 the general results of Chapter 3 are furnished into explicit descriptions of a wide variety of examples of geometries covered by such methods. In particular, the authors thoroughly discuss the geometries corresponding to $$|1|$$-gradings, and the parabolic contact geometries, which have an underlying contact structure.
In Chapter 5 the notion of Weyl structures is used to associate to any parabolic geometry a class of distinguished connections and to define classes of distinguished curves. Also, 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, a more explicit description of the canonical Cartan connection is obtained.
The first parts of the book (Chapters 1 and 2) provide the necessary background and motivation. Chapter 1 contains a survey of Cartan’s geometries and is particularly useful for a general differential geometry audience. This chapter also contains an elementary treatment of conformal (pseudo)-Riemannian structures. Chapter 2 contains background material on semisimple Lie algebras and Lie groups. It also discusses the structure theory and representation theory of real semisimple Lie algebras, a theory typically scattered in textbooks, but rather difficult to learn quickly elsewhere.

##### MSC:
 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 52C15 Packing and covering in $$2$$ dimensions (aspects of discrete geometry) 53A40 Other special differential geometries 53B15 Other connections 53C05 Connections (general theory) 58A32 Natural bundles 53A55 Differential invariants (local theory), geometric objects 53C10 $$G$$-structures 53C30 Differential geometry of homogeneous manifolds 53D10 Contact manifolds (general theory) 58J70 Invariance and symmetry properties for PDEs on manifolds