Introduction to geometry (Q2782057)

The book under review is a welcome introduction in a modern manner to some of the very basic branches of geometry, namely affine, Euclidean, and projective geometry. The final chapter then explains a number of essential ideas involved in the Erlangen Program, thus providing a unifying view of the aforementioned geometries.NEWLINENEWLINETo describe the contents of the book in some detail, we mention that the first chapter exposes the needed notions from linear algebra, with a special emphasis upon the basic theorems on bilinear and quadratic forms, and orthogonal and symmetric linear transforms as well. Several particularly interesting topics are outlined in the exercises at the end of the chapter. Among these, one could mention the canonical form of symplectic bilinear forms, or the basic properties of the Pauli matrices.NEWLINENEWLINEChapter~2 has the title ``Affine and Euclidean geometry'' and starts with a funny motto taken from a letter of W.~Bolyai to his son J.~Bolyai. (The father warns his son against approaching the so-called parallels problem. Fortunately, the son didn't follow father's warning, and eventually became one of the founding fathers of the non-Euclidean geometries!) The approach to affine and Euclidean geometries heavily leans on the linear algebra developed in Chapter~1. The first part of Chapter~2 concerns affine spaces and subspaces, affine parallelism, a few classical results, e.g., Desargues' and Pappus' theorems, and ends with a study of the affine group. The second part of Chapter~2 is mainly devoted to the Euclidean geometry: Euclidean affine spaces, orthogonal linear manifolds, isometry groups. The chapter ends with the classifications of hyperquadrics, firstly from the affine point of view, and then from the metric one. Again we find several noteworthy facts outlined in the exercises at the end of the chapter, e.g., Helly's theorem.NEWLINENEWLINEChapter~3 is devoted to projective geometry. Projective spaces, subspaces, morphisms and groups are firstly introduced. A large part of the chapter then concerns the proof of the fact that, roughly speaking, the ``geometric'' projective spaces are ``algebraic'' projective spaces. The corresponding reasonings are carried out in detail. Then the authors prove the fundamental theorem of projective geometry. In the second part of the chapter, the projective hyperquadrics play a prominent role. Several related classical geometric topics are also approached, e.g., polar pairs or biraport theory. The chapter ends with a section on algebraic plane curves. It is needless to mention again that a lot of interesting exercises can be found at the very end of the chapter.NEWLINENEWLINEThe concluding chapter of the book is a very short one and is intended to discuss, in an informal manner, some of the ideas involved in the Erlangen Program, to the extent necessary to provide a unifying view of the whole book.NEWLINENEWLINETo conclude with, we briefly mention some essential features of the textbook under review: It is a clearly written and very informative textbook. A lot of historical notes are spread throughout the text. It is one of those books showing that it is possible (though certainly not easy) to write mathematics in a good literary style. Even though intended as a textbook for first year graduate students, the book deserves to be read by everyone.