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Projective and Cayley-Klein geometries. (English) Zbl 1107.51001

Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-35644-4/hbk). xvi, 432 p. (2006).
This monograph will likely be, for the foreseeable future, the reference for the algebraic approach to multidimensional projective geometry as well as the metric geometries living therein. It contains material that can be found in a similar presentation in [W. Burau, Mehrdimensionale projektive und höhere Geometrie. (Mathematische Monographien. 5.) Berlin: VEB Deutscher Verlag der Wissenschaften (1961; Zbl 0098.34001), E. Artin, Geometric algebra. Wiley Classics Library. New York etc.: John Wiley & Sons Ltd./Interscience Publishers, Inc. (1988; Zbl 0642.51001) and R. Baer, Linear algebra and projective geometry. Pure and Applied Mathematics, 2., New York: Academic Press Inc., Publishers VIII (1952; Zbl 0049.38103)], but the scope of the present book is wider than that of its predecessors. It is divided into two parts, one studying finite-dimensional projective spaces over skew fields, including collineations, cross-ratios, projective maps, duality, correlations, polarities, quadrics, null systems, line complexes. Among proper skew fields, the quaternions receive special attention.
The second part, devoted to Cayley-Klein geometries, starts with a presentation of the classical groups and of vector spaces with a scalar product, these two being the ingredients needed for defining geometries in projective spaces. Orthogonal geometries, reflections, spherical and ellptic geometry, hyperbolic and Möbius geometry, and projective symplectic geometry are treated in great detail, with many more figures than in all of its predecessors. A final section on transformation groups gives a higher-level perspective on these geometries, in the spirit of Klein’s Erlanger Programm.
There are 114 bibliographic entries. In the reviewer’s opinion, [E. M. Schröder, Vorlesungen über Geometrie. Band 3: Metrische Geometrie. Mannheim etc.: BI-Wiss.-Verlag (1992; Zbl 0754.51004)] should be added to this book’s list of references.

MSC:

51-02 Research exposition (monographs, survey articles) pertaining to geometry
51A05 General theory of linear incidence geometry and projective geometries
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51N15 Projective analytic geometry
51F20 Congruence and orthogonality in metric geometry
14N05 Projective techniques in algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
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