Geometries (Q2881014)

Based on an introductory course delivered at the Independent University of Moscow in 2003 and 2006, this is a leisurely survey of the models of classical geometries over the real numbers, as well as of some elementary finite geometries, with a group-theoretical and a mild category-theoretical flavor. Although some familiarity with Euclidean geometry is assumed, some basic facts of Euclidean geometry are gathered in a Chapter 0. Chapter 1, Toy Geometries and Main Definitions, is devoted to the groups of symmetries of the square, the cube, the circle, the sphere, an introduction to transformation groups and to the category of geometries. Chapters 2 through 5 are devoted to groups, first in the abstract setting, with a definition of group presentations, then moving on to more geometrically flavored topics, such as finite subgroups of \(SO(3)\), discrete subgroups of the Euclidean isometry group, with frieze, wallpaper, and crystallographic groups surveyed, Coxeter groups and their associated geometries. Chapter 6 is on spherical geometry, whereas Chapters 7 through 10 are devoted to hyperbolic geometry in three guises, as Poincaré disk and half-plane models and as Cayley-Klein model, with a proof that they are all isomorphic. The remaining chapters present a short history of non-Euclidean geometry up to 1899, projective geometry over the reals, a chapter showing that Cayley's ``projective geometry is all geometry'' applies to the geometries considered earlier, one on finite geometries, one on the hierarchy of the geometries discussed in the text, one on morphisms of geometries, which leaves the area of elementary geometric discourse to explore covering spaces, \(G\)-bundles, Lie groups, geometric vector bundles. There are two appendices, one containing the axioms and the propositions of Euclid's \textit{Elements}, Book I, and the other one presenting Hilbert's axioms.NEWLINENEWLINEThe author's methodological point of view for presenting geometries via their models (usually over the real numbers) and their groups of transformations, is that the author believes (and he recognizes this belief as a bias, albeit a very common one today) that the axiomatic approach is ``hopelessly outdated and no longer belongs to contemporary mathematics'' (p.\ xiv). The author would have been right if the axiomatic approach consisted only of the works he mentions, all prior to 1900. Had the author have been aware that the 20th century brought us [\textit{F. Bachmann}, Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. ergänzte Aufl. . Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0254.50001)], whose first edition was translated into Russian in 1969, he would have noticed that his problems 6.11 and 6.12, asking to prove that the medians and the altitudes of a spherical triangle are concurrent, allow for a proof that truly reveals the geometrical reason why those lines are concurrent, based on very weak assumptions, that would have simultaneously solved the problems for the hyperbolic and the Euclidean settings, rather than being the result of tedious computations in algebraically-presented models, and thus much closer in spirit to what we are told on page 7, namely that ``the coordinate approach to geometry [\(\ldots\)] is an ugly caricature of what Euclidean geometry really is.''