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Projective and Cayley-Klein Geometries

A. L. Onishchik and R. Sulanke
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Gizem Karaali
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This book is a comprehensive account of projective geometry and other classical geometries, starting from the basics of linear algebra and Euclidean and affine geometry, and exhaustively covering all the details that anyone could ever ask for. It is well-written and the many exercises and many figures (some in color!) make it a very usable text.

However, the readers of the MAA Reviews may need to know that it is probably not suitable for use as a textbook in the undergraduate classroom. The authors themselves assert in their introduction that they intended to write "a manual and a workbook, … and by no means … the text on which a lecture should be based." Overall, the text is a bit too technical for a general audience, and the lack of historical background, though also mentioned in the introduction, makes the text more inaccessible to a regular person passing by who wants to learn some projective geometry through a casual read. You have to be serious when you read this book, you have to want to learn projective geometry very well, and you have to work for it.

Understandably the authors wish to keep the text at a reasonable length. To this end, they intentionally avoid most background material, be it historical or mathematical. Quite regularly they refer the reader to their own two volumes [1,2] for such material. Now to the best of my knowledge, [1] and [2] have not yet been translated into English. In addition, even for readers fluent in German and with access to both [1] and [2], it is not always convenient to be so regularly expected to go to them for some basic lemmas, or an understanding of the Erlangen Programm which is essential to the main agenda of the whole book. Otherwise, the book is mostly self contained and very systematic in its development of projective geometry and the other classical geometries it studies.

My proposed audience for this book coincides with the publisher's advice: graduate students and researchers in mathematics will find this book most useful if they wish to gain a complete understanding of the ideas of the various classical geometries or need a reference for all the necessary technical details without any relation to the historical development of the subject. For these readers, the book is a jewel long yearned for, and finally found. However, the mere mortals who want something more digestible will need to look somewhere else.


[1] A. L. Onishchik, R. Sulanke, Algebra und Geometrie I, VEB Deutscher Verlag der Wissenchaften, Berlin, 1986. 2nd edition.

[2] A. L. Onishchik, R. Sulanke, Algebra und Geometrie II, VEB Deutscher Verlag der Wissenchaften, Berlin, 1988.

Gizem Karaali is assistant professor of Mathematics at Pomona College. She enjoys experiencing the many ways that geometry stretches the brain.

 Projective Geometry.- Cayley-Klein Geometries.- Appendix.- Bibliography.- Index.