Geometry. I, II. Transl. from the French by M. Cole and S. Levy. (English) Zbl 0606.51001

Berger, Marcel

Universitext. Berlin etc.: Springer-Verlag. I: XIII, 428 p.; DM 74.00; II: X, 406 p.; DM 74.00 (1987).

This two-volume textbook is the English translation of the five-volume French original [Vol. 1. (1977; Zbl 0382.51011); Vol. 2. (1977; Zbl 0382.51012); Vol. 3. (1978; Zbl 0423.51001); Vol. 4. (1978; Zbl 0423.51002); Vol. 5. (1977; Zbl 0423.51003)].
The contents include the main fields of elementary geometry, and of convex geometry, e.g. affine and projective geometry, Euclidean spaces, triangles, spheres, circles, convex sets, polytopes, projective and affine quadrics, projective and Euclidean conics, elliptic and hyperbolic geometry, and spherical geometry.
In addition, the author deals with special topics in convex geometry and Euclidean space, e.g. the classification of regular polytopes, the isoperimetric inequality, the theorem of Helly and Krasnosel’skij, Poncelet’s theorem, and Cauchy’s theorem. These results are often absent from comperable books.
As the author points out, his main aim is the motivation and illustration of given definitions, notations and results. This kind of illustration is highly recommendable and specific for this textbook. The book is a rich source for exercises (most of them more difficult than those in comparable books. See the companion volume of problems and solutions by the author, P. Pansu, J.-P. Berry, and X. Saint- Raymond, ”Problems in Geometry” (1984; Zbl 0543.51001), French original in (1982; Zbl 0489.51001)), for problems and its extensive bibliography.

Reviewer: G.Ehrig

MSC:

51-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
52-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry
51M05	Euclidean geometries (general) and generalizations
51M10	Hyperbolic and elliptic geometries (general) and generalizations
51N10	Affine analytic geometry
51N15	Projective analytic geometry
51N20	Euclidean analytic geometry
52Bxx	Polytopes and polyhedra
52A40	Inequalities and extremum problems involving convexity in convex geometry
52A35	Helly-type theorems and geometric transversal theory

Keywords:

crystallographic groups; orthogonal group; Cauchy’s theorem on the rigidity of convex polyhedra; theorems of Witt and Cartan-Dieudonné; polygonal billards; Villarceau circles on the torus; Clifford parallelism; textbook; elementary geometry; convex geometry; affine and projective geometry; Euclidean spaces; triangles; spheres; convex sets; polytopes; projective and affine quadrics; projective and Euclidean conics; elliptic and hyperbolic geometry; spherical geometry; classification of regular polytopes; isoperimetric inequality; theorem of Helly and Krasnosel’skij; Poncelet’s theorem; bibliography

Citations:

Zbl 0382.51011; Zbl 0382.51012; Zbl 0423.51001; Zbl 0423.51002; Zbl 0423.51003; Zbl 0543.51001; Zbl 0489.51001