Cambridge: Cambridge University Press. xv, 333 p. £ 24.95, $ 35.95 (pbk); £ 70.00, $ 100.00 (hbk) (2001).

This is not just another book on the local differential geometry of curves and surfaces: there are numerous textbooks on the topic but most modern textbooks give curves and surfaces only a very elementary treatment and leave concealed the richness of the theory, that becomes apparent when studying, for example, the classical textbooks of {\it G. Darboux} [Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Vols. I--IV; Reprint, Chelsea, New York (1972;

Zbl 0257.53001)], {\it L. Bianchi} [Lezioni di geometria differenziale; Enrico Spoerri, Pisa (1923;

JFM 49.0498.06)], or {\it W. Blaschke} [Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. I--III; Springer, Berlin (1930;

JFM 56.0588.06), (1923;

JFM 49.0499.01), and (1929;

JFM 55.0422.01)]. The book under review is different. It touches upon a variety of interesting topics that are usually not to be found in textbooks, in particular, different kinds of singularities that naturally occur, for example, from certain constructions are discussed in detail. Also, the book leads the reader to current research results, by the author and his colleagues and by the Russian master V. I. Arnol’d of singularity theory and his school. This second edition of {\it I. R. Porteous}’ book extends the first edition (1994;

Zbl 0806.53001) by the two chapters 17 and 18 on flexcords of surfaces and on curves and surfaces in the $2$- and $3$-sphere.
Nevertheless the book certainly qualifies as a textbook, suitable for a course on curves and surfaces, as any material needed beyond a minimum background in analysis and linear algebra is discussed. Various “real world” examples are included that may motivate a student or make the material more accessible.
The following details shall illustrate the above remarks:
Chapter 1, “Plane curves”, deals with (parametrized) curves in the Euclidean plane. The notion of curvature is introduced via contact with a (parametrized) circle. Regular and unit speed curves are only treated as special cases. Much attention is paid to the evolute (focal curve) -- evolvente relationship between curves, the parallel curves of a given curve are studied, and the various kinds of singularities that can occur in these constructions are discussed in detail.
After some basic facts on (bi-) linear algebra, polynomials, projective, and inversive geometry have been collected in Chapter 2, the ideas developed in the first chapter are applied to “Plane kinematics” in Chapter 3: the inflection circle, ball point, cubic of stationary curvature, and the Burmester points of a rigid motion -- all being described in terms of special points of the orbits of the motion -- are discussed. The rolling of a curve on another curve, previously touched upon in the context of the evolventes of a curve, is taken up again. In the last section “Caustics” a connection to optics is drawn.
The following Chapter 4, “The derivative of a map”, provides the usual basics from multidimensional analysis that is needed in differential geometry.
In Chapter 5, “Curves on the unit sphere”, some of the ideas from the first chapters are carried over to the $2$-sphere: the geodesic curvature of a spherical curve is introduced, and some aspects of spherical kinematics are discussed, drawing a connection to geology (plate tectonics).
Chapter 6, “Space curves”, does indeed not only deal with space curves: before getting to the Frenet formula for a space curve and to the notion of parallel curves, the tangent developable of a space curve, the (space) evolute, and the focal surface (the tangent developable of the evolute) are discussed. Again, regularity questions and the occuring singularities are investigated.
The following three chapters provide technical background: in Chapter 7, “$k$-times linear forms”, some notions related to multilinear maps, quadratic, and cubic forms are introduced and the “Use of complex numbers” in this context, relevant to the study of surfaces, is explained; in Chapter 8, “Probes”, a technique to investigate singularities of maps between vector spaces is introduced. The next short Chapter 9, “Contact”, explains the notion of contact equivalence, which provides an application for the material from the previous chapter.
Chapter 10, “Surfaces in $R^3$”, starts with a comprehensive introduction that outlines what will be discussed in the following chapters, in particular, concerning the relation between a surface and its focal surface (evolute). After this introduction the usual material is covered: the first and second fundamental forms and the principal curvatures are discussed. Considering the corresponding centres of curvature one is led to the evolute of a surface, and first relations between the surface and its focal surface are discussed; in particular it is shown that the curvature lines of the surface yield geodesics of the evolute where it is regular.
Chapter 11, “Ridges and Ribs”, then discusses the various types of singularities that the focal surface of a regular surface can have. An important idea is to consider the “normal focal surface” as a surface in the tangent bundle of Euclidean $3$-space. Chapter 12, “Umbilics”, continues the investigation of isolated umbilics begun in the preceding chapter. In particular the shape of the principal curvature lines around an umbilic is discussed, and ways to classify umbilics are presented. Here the discussions are more colloquial, a short section on the index of an umbilic collects some facts without proofs -- however, many references to classical and modern literature are given where the reader may find more detailed information. Remarks on parallel surfaces and on the effect of deformation of a surface on umbilics conclude the chapter.
Chapter 13, “The parabolic line”, is concerned with the locus of parabolic points (points of Gaussian curvature $0$) of a surface. It starts with a proof of Gauss’ theorema egregium, and then turns to a description of the singularities of the Gauss map at parabolic points. Koenderink’s theorems on the relation between the Gaussian curvature of a surface and its contour under a parallel or central projection, respectively, are presented and the consequences for parabolic points are investigated -- thus establishing a connection to shape recognition. Finally the subparabolic lines of a regular surface are studied: these are the lines that correspond to parabolic lines on the focal surface; in particular, their behaviour at umbilics is related to that of the curvature lines of the surface.
The reconstruction of a surface from (one sheet of) its evolute is the theme of Chapter 14, “Involutes of geodesic foliations”: the starting point here is the earlier observation that the curvature lines of a surface yield geodesics on the focal surface; the reconstruction then proceeds similar to the case of planar curves. However, assuming the evolute to be regular its involute may develop certain types of singularities that are discussed in detail. Finally, Coxeter groups are brought into play and it is outlined how they give rise to singularities of curves or surfaces -- at this point the labeling of singularities in the book under review is explained.
Chapter 15, “The circles of a surface”, deals with osculating circles of a surface. At the basis of the discussions is Meusnier’s classical theorem. As a refinement of this theorem the existence of higher order osculating circles in a given direction is related to various types of points of the surface discussed earlier throughout the book, in particular, to umbilics. In case the Meusnier sphere becomes the tangent plane of the surface, the geometry of its intersection curve with the surface is examined. Using, as explained at the end of Chapter 13, an inversion to flatten the Meusnier sphere these investigations carry over to the Meusnier sphere.
In Chapter 16, “Examples of surfaces”, the theory presented in the preceding chapters is exemplified: among the discussed surfaces are the general ellipsoid, bumpy spheres that emanate from perturbing the equation of the unit sphere by a (small) third order term, and the minimal monkey saddle.
Chapter 17, “Flexcords of surfaces”, is the first of two new chapters: it takes up the discussion of subparabolic lines (now called flexcords) from Chapter 13 and presents recent developments on the subject. In particular, flexcord points of implicitly given surfaces are characterized; and the “birth” or “death” of umbilics, touched upon earlier in Chapter 12, is discussed in more detail -- bumpy spheres provide an interesting example.
The second new Chapter 18, “Duality”, deals with Arnol’d’s notions of the dual and the derivative curve of an (oriented) curve in the $2$-sphere, thus complementing material from Chapter 5, and with generalizations of these notions to surfaces and curves in the $3$-sphere. Singularities that occur from these constructions are examined, and implications for the respective evolutes are discussed.
In each chapter a set of exercises is formulated.
The reviewer found this an interesting and inspiring book, a slight drawback only being the fact that definitions are incorporated into the text which makes it harder to locate them -- as a consequence the reviewer failed in a first attempt to start reading the book in the middle. Many instructive illustrations can help the reader to understand the discussed notions and facts, and many interesting historical remarks and references are given throughout the text.