##
**Classical geometries in modern contexts. Geometry of real inner product spaces.**
*(English)*
Zbl 1084.51001

Basel: Birkhäuser (ISBN 3-7643-7371-7/hbk). xii, 242 p. (2005).

The volume under review can be seen as the third part of an opus magnum, the first two being [Geometrische Transformationen (B. I. Wissenschaftsverlag, Mannheim) (1992; Zbl 0754.51005)] and [Real geometries (B. I. Wissenschaftsverlag, Mannheim) (1994; Zbl 0819.51002)], in which the author’s aim is to present (i) models of classical geometries in a dimension-free setting, (ii) the relevance of functional equations in geometry, (iii) very general forms of characterizations of mappings under mild hypothesis (of the Mazur-Ulam, Alexandrov-Zeeman, Beckman-Quarles type), (iv) a very precise conceptual framework for Klein’s Erlangen programme, with constant emphasis on the interplay between geometries and groups of transformations.

In Chapter 1, the author introduces the notion of a translation group associated to a real inner product space \(X\), defines Euclidean and hyperbolic geometry over \(X\) (in the case of hyperbolic geometry, we are presented with the Weierstrass-inspired model put forward by the author in [N. K. Artémiadis (ed.) et al., Proceedings of the 4th international congress of geometry, Thessaloniki, Greece, May 26–June 1, 1996. Athens: Aristotle University of Thessaloniki. 12–21 (1997; Zbl 0888.51020)], and presented in his textbook [Ebene Geometrie, Spektrum, Heidelberg (1997; Zbl 0870.51001)]), and proves an astonishing characterization, of a kind that brought the result of H.-C. Wang [Pac. J. Math. 1, 473–480 (1951; Zbl 0044.19602)] to the reviewer’s mind, of the Euclidean and hyperbolic geometries as the only geometries over \(X\) satisfying rather general conditions involving the existence of a function that has only some of the properties the metric would have in geometries with a mildly special kind of translation group (first published by the author in [Publ. Math. 63, 495–510 (2003; Zbl 1052.39022)]).

Chapter 2 is devoted to an in-depth analysis of the models of Euclidean and hyperbolic geometry that are defined in the previous chapter, the emphasis being naturally on the hyperbolic case. Among the topics: determining the equations of lines, as defined in three different ways (by Blumenthal, Menger, and the author), hyperplanes, subspaces, equidistant surfaces, ends, parallelism, angles of parallelism, horocycles, the Cayley-Klein model, various characterizations of isometries and of translations.

Chapter 3 is devoted to the sphere geometries of Möbius (first presented in [Aequationes Math. 66, 284–320 (2003; Zbl 1081.51500)]) and Lie, in which we encounter Poincaré’s model of hyperbolic geometry, the main notions of these geometries (such as Lie and Laguerre cycles, Lie and Laguerre transformations), with their remarkable characterization under the very “mild hypothesis” that they are bijections preserving the cycle contact relation (but not necessarily the negation of this relation), which was the subject of the author’s [Result. Math. 40, 9–36 (2001; Zbl 0995.51003)].

Chapter 4 looks at space-times and their groups of transformations, Minkowski space-time and Lorentz transformations, de Sitter’s world, Einstein’s cylinder world, again with many characterizations of the corresponding transformations under mild hypotheses, as well as a characterization of a general notion of Lorentz-Minkowski distance, and a theorem that uncovers the very strong connection existing between hyperbolic motions and Lorentz transformations. The author’s model of hyperbolic geometry may thus be said to be the unifying thread, and the various results proved in this book which rely very heavily on this model testify to the significance of the discovery of this model, which turns out to be not “just another model of hyperbolic geometry”, but one that allows, by the very fact that its point-set is that of Euclidean geometry or of Minkowski space-time, a fruitful comparison.

The mathematical prerequisites are minimal – the rudiments of linear algebra suffice – and all theorems are proved in detail. Following the proofs does not involve more than following the lines of a computation, and the author makes every effort to avoid referring to a synthetic geometric understanding, given that he aims at attracting readers with a distaste for synthetic geometry, which, given the academic curricula of the past decades, represent the overwhelming majority of potential readers of any mathematical monograph. One of the lessons of this monograph is that there is a coordinate-free analytic geometry, which significantly simplifies computations and frees the mind from redundant assumptions. There is only minimal overlap with the first two volumes, the aim of the few repetitions being that of ensuring the volume’s independent readability.

In the realm of synthetic (axiomatic) geometry, the dimension-free approach can be traced back to H. N. Gupta [Contributions to the axiomatic foundations of Euclidean geometry (Ph. D. Thesis, University of California, Berkeley) (1965)] and W. Schwabhäuser [J. Reine Angew. Math. 242, 134–147 (1970; Zbl 0199.55002)].

In Chapter 1, the author introduces the notion of a translation group associated to a real inner product space \(X\), defines Euclidean and hyperbolic geometry over \(X\) (in the case of hyperbolic geometry, we are presented with the Weierstrass-inspired model put forward by the author in [N. K. Artémiadis (ed.) et al., Proceedings of the 4th international congress of geometry, Thessaloniki, Greece, May 26–June 1, 1996. Athens: Aristotle University of Thessaloniki. 12–21 (1997; Zbl 0888.51020)], and presented in his textbook [Ebene Geometrie, Spektrum, Heidelberg (1997; Zbl 0870.51001)]), and proves an astonishing characterization, of a kind that brought the result of H.-C. Wang [Pac. J. Math. 1, 473–480 (1951; Zbl 0044.19602)] to the reviewer’s mind, of the Euclidean and hyperbolic geometries as the only geometries over \(X\) satisfying rather general conditions involving the existence of a function that has only some of the properties the metric would have in geometries with a mildly special kind of translation group (first published by the author in [Publ. Math. 63, 495–510 (2003; Zbl 1052.39022)]).

Chapter 2 is devoted to an in-depth analysis of the models of Euclidean and hyperbolic geometry that are defined in the previous chapter, the emphasis being naturally on the hyperbolic case. Among the topics: determining the equations of lines, as defined in three different ways (by Blumenthal, Menger, and the author), hyperplanes, subspaces, equidistant surfaces, ends, parallelism, angles of parallelism, horocycles, the Cayley-Klein model, various characterizations of isometries and of translations.

Chapter 3 is devoted to the sphere geometries of Möbius (first presented in [Aequationes Math. 66, 284–320 (2003; Zbl 1081.51500)]) and Lie, in which we encounter Poincaré’s model of hyperbolic geometry, the main notions of these geometries (such as Lie and Laguerre cycles, Lie and Laguerre transformations), with their remarkable characterization under the very “mild hypothesis” that they are bijections preserving the cycle contact relation (but not necessarily the negation of this relation), which was the subject of the author’s [Result. Math. 40, 9–36 (2001; Zbl 0995.51003)].

Chapter 4 looks at space-times and their groups of transformations, Minkowski space-time and Lorentz transformations, de Sitter’s world, Einstein’s cylinder world, again with many characterizations of the corresponding transformations under mild hypotheses, as well as a characterization of a general notion of Lorentz-Minkowski distance, and a theorem that uncovers the very strong connection existing between hyperbolic motions and Lorentz transformations. The author’s model of hyperbolic geometry may thus be said to be the unifying thread, and the various results proved in this book which rely very heavily on this model testify to the significance of the discovery of this model, which turns out to be not “just another model of hyperbolic geometry”, but one that allows, by the very fact that its point-set is that of Euclidean geometry or of Minkowski space-time, a fruitful comparison.

The mathematical prerequisites are minimal – the rudiments of linear algebra suffice – and all theorems are proved in detail. Following the proofs does not involve more than following the lines of a computation, and the author makes every effort to avoid referring to a synthetic geometric understanding, given that he aims at attracting readers with a distaste for synthetic geometry, which, given the academic curricula of the past decades, represent the overwhelming majority of potential readers of any mathematical monograph. One of the lessons of this monograph is that there is a coordinate-free analytic geometry, which significantly simplifies computations and frees the mind from redundant assumptions. There is only minimal overlap with the first two volumes, the aim of the few repetitions being that of ensuring the volume’s independent readability.

In the realm of synthetic (axiomatic) geometry, the dimension-free approach can be traced back to H. N. Gupta [Contributions to the axiomatic foundations of Euclidean geometry (Ph. D. Thesis, University of California, Berkeley) (1965)] and W. Schwabhäuser [J. Reine Angew. Math. 242, 134–147 (1970; Zbl 0199.55002)].

Reviewer: Victor V. Pambuccian (Phoenix)

### MSC:

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

51B10 | Möbius geometries |

51B25 | Lie geometries in nonlinear incidence geometry |

51M05 | Euclidean geometries (general) and generalizations |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

83A05 | Special relativity |

83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |

39B52 | Functional equations for functions with more general domains and/or ranges |