Synopses & Reviews
The first part of the book presents n-dimensional projective geometry over an arbitrary skew field; the real, the complex, and the quaternionic geometries are the central topics, finite geometries playing only a minor part. A detailed proof of the main theorem of projective geometry is followed by discussions concerning the cross ratio, Staudt's main theorem, duality, correlations, quadrics, and null systems. Polarities and null systems are classified for these geometries. Finally, changes of the geometric as well as the algebraic structures resulting from restrictions and extensions of the scalar domain are described, in particular, this includes the Hopf fibrations. The second part deals with the classical linear and projective groups and the associated geometries; it is based on the classification of polarities. The guiding principle for this is provided by F. Klein's Erlangen Program. The theory of vector spaces with scalar product, probably going back to E. Artin, is studied in detail including the theorem of E. Witt. After a general investigation of the projective geometry corresponding to a polarity, the elementary spherical, elliptic, and hyperbolic geometries are presented. For them complete systems of invariants for pairs of subspaces are established: stationary angles and distances. Moreover, based on the study of the associated symmetric linear endomorphisms, the symmetric bilinear forms and quadrics are classified for these geometries. Further topics of the book are Mvbius geometry as well as elementary symplectic projective geometry. The last section contains a summary of selected results and problems from the geometry of transformation groups; in particular, theclassification results for transitive actions of Lie group on spheres and projective spaces are described. The appendix collects brief accounts of some fundamental notions from algebra and topology with corresponding references to the literature. The book is intended to be a self-contained introduction into projective geometry for students and others interested in this subject.
Review
From the reviews: "This book is a comprehensive account of projective geometry and other classical geometries ... exhaustively covering all the details that anyone could ever ask for. It is well-written and the many exercises and many figures ... make it a very usable text. ... My proposed audience for this book coincides with the publisher's advice: graduate students and researchers in mathematics will find this book most useful ... . For these readers, the book is a jewel long yearned for, and finally found." (Gizem Karaali, MathDL, November, 2007)
Synopsis
Projective geometry, and the Cayley-Klein geometries embedded into it, were originated in the 19th century. It is one of the foundations of algebraic geometry and has many applications to differential geometry. The book presents a systematic introduction to projective geometry as based on the notion of vector space, which is the central topic of the first chapter. The second chapter covers the most important classical geometries which are systematically developed following the principle founded by Cayley and Klein, which rely on distinguishing an absolute and then studying the resulting invariants of geometric objects. An appendix collects brief accounts of some fundamental notions from algebra and topology with corresponding references to the literature. This self-contained introduction is a must for students, lecturers and researchers interested in projective geometry.
Synopsis
This book offers an introduction into projective geometry. The first part presents n-dimensional projective geometry over an arbitrary skew field; the real, the complex, and the quaternionic geometries are the central topics, finite geometries playing only a minor part. The second deals with classical linear and projective groups and the associated geometries. The final section summarizes selected results and problems from the geometry of transformation groups.
Table of Contents
Projective Geometry.- Cayley-Klein Geometries.- Appendix.- Bibliography.- Index.