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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.

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