Geometries on surfaces (Q2778901)

Many familiar geometries like the real hyperbolic or the complex affine plane or the geometry of plane sections of a Euclidean sphere are topological geometries in the sense that the point set is a topological space (often even a manifold) and there exists a topology for the set of lines or circles rendering the relevant geometric operations continuous. Because of the particular topological properties of low-dimensional locally compact spaces, the problems in the case of geometries living on surfaces (two-dimensional topological manifolds) are very different from those arising for point sets of higher dimension. This has become quite obvious in the book \textit{Compact Projective Planes} [CP] by the Ref. et al. [W. de Gruyter (1995; Zbl 0851.51003)]. (Note that `dimension' always refers to the covering dimension of a topological space and not to the rank of the underlying incidence structure.)NEWLINENEWLINENEWLINEThe book under review is an excellent companion to [CP]. The two texts have only one area in common: projective planes with point set and line set homeomorphic to the point space \(\mathbb P\) of the real projective plane. Here, the authors discuss the results obtained in [CP], but they do not repeat the proofs. Otherwise, the two books do not overlap.NEWLINENEWLINENEWLINE\textit{Geometries on surfaces} begins with a short informal chapter introducing different kinds of geometries to be dealt with in the main body of the text. The most prominent of these are either linear spaces (any \(2\) distinct points are on a unique line) or circle planes (any \(3\) pairwise non-parallel points are on a unique circle). If the point space of such a geometry is a surface and if all lines or circles are connected \(1\)-manifolds (homeomorphic to \(\mathbb R\) or \(\mathbb S\)), then the geometry is said to be \textit{flat}. The authors consider only topological geometries. A linear space is called \textit{stable}, if join and intersection are continuous and the set of pairs of intersecting lines is open. The study of the interplay between the geometrical and the topological structure is an important and very attractive feature of the book. In the case of flat geometries, the two structures seem to be even more tightly interwoven than in higher dimensions, exemplified in the following. NEWLINENEWLINENEWLINEChapter 2 is devoted to flat linear spaces and comprises one fourth of the total text. By a result of \textit{R. Löwen} [Geom. Dedicata 58, 175-183 (1995; Zbl 0845.51012)], only \(3\) surfaces can carry a flat stable plane: the compact space \(\mathbb P\), the Möbius strip \({\mathbb P} {\smallsetminus} \{p\}\), and a disk (or \({\mathbb R}^2\)). In the first two cases the line space is homeomorphic to \(\mathbb P\), in the last one it is a Möbius strip. Conversely, each flat linear space which is realized on a disk or on \(\mathbb P\) is in fact a stable plane. The proof follows from a theory of convexity, which is developed for linear spaces on a disk. Flat stable planes on the \(3\) surfaces are discussed in detail. Basic results are presented with full proofs (collineations are continuous and form a Lie group of dimension at most~\(8\)), others are just stated and commented upon (with references for their proofs). All flat stable planes with a group of dimension at least \(3\) are known explicitly. On each of the \(3\) surfaces there is an abundance of less homogeneous geometries. The authors give much room to the construction and description of various kinds of examples, from the Moulton planes, which were discovered a century ago and are characterized by the existence of a \(4\)-dimensional collineation group, to rigid planes admitting no collineation other than the identity. Each convex open domain in a surface geometry yields an \({\mathbb R}^2\)-plane. Such geometries may be glued together along a suitable curve to form a new surface geometry. Several special topics (differentiable planes, groups of projectivities, pseudoline arrangements, etc.) are only shortly touched upon.NEWLINENEWLINENEWLINEThe next \(3\) chapters (about one half of the book and the main part of it) concern circle planes. The classical (Miquelian) models are given by the plane sections of a sphere or a cylinder in \({\mathbb R}^3\) and of a ruled quadric in projective \(3\)-space, the latter surface being homeomorphic to a torus \({\mathbb T}^2\). On all \(3\) surfaces rather arbitrary families of Jordan curves may be used instead of the ordinary circles to form new flat circle planes. Any automorphism of the underlying incidence geometry is continuous, i.e., it is an automorphism of the topological circle plane. If \(p\) is a point of the circle plane \({\mathcal C}\), then the `derived' linear space \({\mathcal C}_p\) has as point set all points not parallel to~\(p\), lines of \({\mathcal C}_p\) consist of the remaining points of the circles through \(p\) and the parallel classes of points not containing \(p\). Spherical, cylindrical, and toroidal circle planes such that derivation always yields an affine plane are known as Möbius, Laguerre, and Minkowski planes respectively. If Miquel's theorem holds locally in one of these planes, then it holds even globally. For all \(3\) types of circle planes there are analogues of the well-known Lenz-Barlotti types of projective planes. Many of these types are realized on surfaces. The more collineations a class of flat circle planes has, the fewer models exist. In particular, a spherical circle plane with a \(4\)-dimensional group or with a point- or circle-transitive group is Miquelian. Again, the authors dedicate a good deal of attention to different descriptions of the classical circle planes and to methods of construction of examples with distinct properties. The plane sections of any ovoid (strictly convex smooth compact surface in \({\mathbb R}^3\)) form a Möbius plane such that all derived planes are classical, and each automorphism of such a Möbius plane extends to a collineation of the surrounding projective space \(\text{PG}(3,{\mathbb R})\). For suitable ovoids the Möbius plane is rigid. Many examples, also rigid ones, can be obtained by `cutting and pasting'. The fairly complicated results of Strambach's classification of spherical circle planes with a \(3\)-dimensional group of automorphisms are presented in their essential aspects but without all details. -- Toroidal and cylindrical circle planes are treated similarly. There exist, however, non-classical flat Minkowski planes with a \(4\)-dimensional group and Laguerre planes with a \(5\)-dimensional group. As before, the classical planes are represented and characterized in several different ways, and construction methods play a major rôle. Large families of examples can be obtained by different kinds of `cutting and pasting'. The more homogeneous Minkowski and Laguerre planes are known and are described explicitly.NEWLINENEWLINENEWLINEChapter 6 deals with compact generalized quadrangles \(\mathcal Q\) whose lines and pencils are homeomorphic to \(\mathbb S\). Either \(\mathcal Q\) or its dual is then anti-regular (any 3 pairwise non-collinear points are joined to \(0\) or exactly \(2\) points). There are close relationships between such anti-regular quadrangles and the \(3\) types of flat circle planes [\textit{A. E. Schroth}, Topological circle planes and topological quadrangles (Harlow, Essex: Longman) (1995; Zbl 0839.51013)]. The~authors summarize Schroth's results. If \(\mathcal Q\) is an anti-regular quadrangle as above, then each derived structure \({\mathcal Q}_p\) is a flat Laguerre plane and \(\mathcal Q\) can be regained as the Lie geometry of \({\mathcal Q}_p\). This sets up a `sisterhood' relation between Laguerre planes \({\mathcal Q}_p\) and \({\mathcal Q}_q\) belonging to the same quadrangle. Similar constructions in the Möbius and the Minkowski case involve in addition a suitable involution. The Apollonius problem as formulated by Schroth asks for the number of circles in a circle plane which are collinear in the corresponding Lie geometry with \(3\) given points. The solutions of this problem for arbitrary flat Möbius, Laguerre, and Minkowski planes are analogous to those in the classical cases. All possibilities are presented in \(3\) tables. NEWLINENEWLINENEWLINEIn the final chapter 7 on `tubular circle planes' the authors interpret aspects of interpolation and approximation theory in the context of circle geometries of higher rank and vice versa. In a tubular circle plane of odd rank \(n\) on the cylinder there is a unique `circle' through any \(n\) pairwise non-parallel points. The affine part of the classical model is given by the polynomials of degree less than \(n\) in the real plane. Many results in the previous chapters have counterparts in this general setting, but the theory is still not as fully developed as in the case \(n=3\). Approximation theory can be utilized to answer geometrical questions; conversely, geometrical arguments may lead to results in approximation theory. NEWLINENEWLINENEWLINEBasic results from other areas are collected in two short appendices, one on metric spaces, manifolds, topological dimension, and fixed points, the other on Lie transformation groups of small dimension (each locally compact, effective transformation group of a surface is a Lie group).NEWLINENEWLINENEWLINEThe main objective of the book, to give an intuitive and fairly complete picture of the wealth of geometries living on surfaces and of the beauty of the subject, has been accomplished in an excellent way. The text provides an easily accessible and well-motivated introduction to topological geometry. On the other hand, essential parts of the book have more the character of a survey, presenting all the facts, but referring to the literature for most of the more complicated proofs.