
Author(s)  Fenchel, W. 

Title  Elementary Geometry in Hyperbolic Space 
Publisher  Walter de Gruyter & Co. 
Year of publication  1989 
Reviewed by  S.V. Rogozin 
The hyperbolic space is wellknown mathematical object. There are many investigations connected with it. The most of them deals with algebraic questions on this structure as well as with many generalizations and applications of this extremely deep idea. However the beginning of the hyperbolic space's study was (besides of highly abstract ground) more utilitarian. In the old articles and books the authors tried to find the most understandable description of this object and to see its connections with other branches of mathematics. One of the directions which was on the interest for majority of mathematicians at the end of the last and at the beginning of this century was the geometry of the hyperbolic space (more precisely elementary geometry of it). But after some time the fashion of the works of such a type has been changed. They became more abstract, practically did not include the full proofs and careful calculations. As a result we can see that oldfashioned objects are not on the disscussion in the modern literature. Anyway these objects still play very important role in the understanding of many new results and their applications. Besides the oldfashioned technics is useful not only for educational purposes but also for many serious and deep mathematical investigations. Taking into account all mentioned the author of the reviewed book, wellknown specialist in different branches of mathematics Werner Fenchel, has seen behind him the goal to gather the most of results on the geometry of hyperbolic space, to introduce them as detailed as possible and from the modern unified point of view. The book seems very interesting combination of the oldfashioned technics and of the deep modernfashioned ideas. Its appearance is in any case the great event in education as well as in science.
The basic model for hyperbolic space using in the book is the wellknown Poincare conformal conotructlon. This model was and still is highly applicable for many results connected with the hyperbolic space. Here it is successfully used for the purpose to understand the geometry in hyperbolic situation. The main attention is given to objects analogous to classical ones, to the relations looking likes the classical ones and so on. Besides are presented quite modern investigations on Kleinian groups which appeared to be useful in the number of discussing questions. So from one side the book is a textbook on geometry of hyperbolic space including an introduction into the several problems of this construction. From the other side it is a reference book about the modern state of some geometrical problems in noneuclidean oase. The latter image is strengthed due to several new results including into the main text and historical notes and remarks ended every section.
The book consists of nine chapters, the short list of references (including only those of books and articles which are connected with the sources of ideas), preface and subject index. In the first two chapters and partly in the third one are on the discussion some terms and notions which can be understood in the Euclidian space. In the rest of the book all notions have the source and sense only as objects of the hyperbolic structure.
In the first chapter ("Preliminaries") are gathered some auxiliary notations and facts needed for the introducing the conformal model. They are quaternions, hyperbolic functions and some relations for them and at last a collection of results for 2x2 complex matrices (trace relations, linear group SL(2,C) and connected with it cross ratio).
The preparations for the construction of the main model are continued in the next chapter ("The Möbius Group"). For this purpose are discussed some properties of the Euclidian spaces E2 and E3, including the short list of results on transfomations on these spaces. The latter leads to the basic idea of this chapter and one of the essential component of the Poincare model. This is the groups of circle and spheres preserving transformations (so called Möbius groups M2 and M3).
In the third chapter ("The Basic Notions of Hyperbolic Geometry") the above mentioned groups are introduced as Möbius groups of upper halfspace. Here are given only elementary objects of hyperbolic geometry (hpoints, hlines, hplanes; general relations between them as well as hyperbolic metrics). So it is constructed the hyperbolic space  Euclidian upper halfspace under Möbius group transformations equipped by hyperbolic metrics.
Then some kind of notions into this space are on the discussion. First of all they are orientationpreserving isometries. Basic facts about them are gathered in the chapter IV ("Isometry Group of Hyperbolic Space"). All the results are interpreted from the geometrical point of view. It gives for instance such purely geometrical objects as spherical and cylindrical surfaces.
The chapter V ("Lines") is again devoted to ("hyperbolic") straight lines. Here the latter are constructed from the coordinate position, but every step of this construction is expressed in terms of matrix presentations. Besides of coordinate discussion of the proper and improper lines (which is in some sense the preparation for the study of polygons in hyperbolicspace) is also presented the idea of pencils and bundles of lines. The latter shows the readers the connection between the hyperbolic space and a projective plane.
Practically all possible questions about polygons are on the discussion in the Chapter VI ("Rightangeled Hexagon"). Some of the results are quite old and known even for Lobachevskii, but many of them are modern.
The calculations and explanations of ideas are given as carefully as possible. Due to this the chapter has the most instructive character in the whole book. But this is a consequence of high applicability of the facts gathered here.
After this the author again returns to the basic objects of geometry (the chapter VII "Points and Planes"). These objects are introduced in the same manner as lines in the chapter V above. Mainly the matrix presentations are studied. The attention is paid also to the projective model, as well as to the pencils and bundles of points and planes.
The last three chapters give us the short introduction to the geometry of surfaces. Obviously these objects are more complicative than the above discussed. So mainly are studied the most simple of the surfaces, spherical onces, until the linear families of them (this is the content of the chapter VIII). Then are defined several different coordinate systems in hyperbolic space (chapter IX "Area and Volume"). This leads to introduction of surface (area) and space (volume) metrics. Due to them are obtained some results on differential geometry in hyperbolic space.
The book will be extremely interesting for all who deals with the problems of hyperbolic space especially for those who works in geometry, topology and complex analysis. Partly this is connected with the wellknown problem of 3dimensional manifolds' classification, because it is appeared that hyperbolic geometry is the key for this problem.
The instructive character of some chapters makes this book very useful for graduate students. It can help them to understand many problems around hyperbolicity.
The evident success of the book deals with the personality of the author  the famous specialist in some branches of mathematics (convex analysis and differential geometry among them).