Geometry. Transl. from the Russian by O. V. Sipacheva. Transl. edited by A. B. Sossinski (Q2716061)

This remarkable book is a serious (and quite successful) attempt to present GEOMETRY as a unified branch of mathematics. And this book represents precisely geometry but not algebraic or analytic technique which fill numerous standard monographs and textbooks of analytical or differential geometry. NEWLINENEWLINENEWLINEThe structure of the book is simple and natural; the authors successively set forth notions and main theorems of different geometries (Euclidean, affine, projective, elliptic, hyperbolic, and others), first, in the plane variant, then in finite-dimensional case; the infinite-dimensional case is presented in the end of the basic part of the book. After the general (and elementary) account of every geometry, the authors present a number of striking and nice geometrical facts related to these geometries.NEWLINENEWLINENEWLINEThe book consists of 6 chapters. Chapter 1 (The Euclidean World) deals with the vector space \(\mathbb{R}^n\), the affine space \(\mathbb{A}^n\) and the Euclidean space \(\mathbb{E}^n\). Among other things, determinants and volumes, simplices and balls are considered. Chapter 2 (The Affine World) deals with the affine space \(\mathbb{A}^n\) and elements of finite-dimensional convex geometry. Chapter 3 (The Projective World) is devoted to a real projective space \(\mathbb{R} P^n\). Chapter 4 (Conics and Quadrics) presents basic geometrical results concerning ellipse, hyperbola and parabola; in addition, the authors touch on some interesting geometrical facts, as the butterfly problem, Pascal's theorem, and others. Chapter 5 (The World of Non-Euclidean Geometries) is devoted to elementary Riemann and Lobachevsky geometry; the chapter also presents the comparison of isometries in the three classical geometries (Euclidean, Riemann, and Lobachevsky). Chapter 6 (The Infinite-Dimensional World) deals with the description of analogues of the most important finite-dimensional geometrical theorems like separation theorems, theorems about subspaces and half-spaces, infinite-dimensional quadrics and so on.NEWLINENEWLINENEWLINEThe book contains addenda with some interesting and more special problems of geometry: I. Geometry and physics (here one can read more about relations between the mathematical and physical notiones of space and the role of Galileo, Newton, Minkowski and Einstein in this problem), II. Polyhedra and polygona (here the authors touch on the classical Euler-Poincaré, Gram-Sommerville, Gauss-Bonnet, Steiner-Minkowski formulas and others), III. Additional questions of projective geometry (here the reader meets the complex projective space \(\mathbb{C} P^n\) and theory of projective duality), IV. Special properties of conics and quadrics (in particular, here Poncelet's theorem and the zigzag theorem are presented), V. Additional topics of non-Euclidean geometries (paving of planes in three classical geometries, axiomatic approach, some models of geometries and so on). The clear and exact description of foundations and basic geometrical results are added with numerous pictures and an excellent set of problems (in the end of the book one can find their solutions, hints, and answers).NEWLINENEWLINENEWLINEOf course, one can find some lacks and deficiencies in any book. To my mind, chapter 6 must be complemented by the description of basic geometrical facts in Hilbert, Banach and even Fréchet spaces, in particular, by beautiful Riesz's lemmas about perpendiculars and almost perpendiculars. Similarly, one can feel the absence of the chapter about manifolds with their remarkable history, related with navigation, astronomy, and geographical maps.NEWLINENEWLINENEWLINEFormally, the book is intended for college undergraduate and graduate students, high school mathematics teachers, and researchers in mathematics and physics. However, I think that this book will be useful in any mathematical library and the acquaintance with it makes each mathematician be happy.