Integral geometry from Buffon to geometers of today
This book presents an exposition of the most important results in the historical development of integral geometry. The origins of this area of mathematics go back to 1777, when Buffon published his \textit{Essai d'arithmétique morale}. There he proves that when a needle is randomly thrown on the boards of a parquet, if the length of the needle is equal to the width of the boards, the probability to lie across two boards is \(\frac{2}{\pi}\). The study of this problem naturally leads to the notion of a measure on the set of all possible positions of the needle. In 1832, Cauchy noticed that the length of a convex curve is proportional to the average of the lengths of its orthogonal projection on all lines through a fixed point, thus establishing what is now called \textit{Cauchy's formula}. Again, a measure on a set of lines is considered. However, it was a serious setback for the developing integral geometry, when Bertrand constructed his paradoxes, with which the author closes the first, introductory chapter.NEWLINENEWLINEIn the second chapter, the author introduces a measure on the set of affine straight lines in the plane and proves the Cauchy-Crofton theorem and a so-called exchange theorem. The first one states that the length of a curve is half of the average number of its intersection points with affine lines; the second one claims that the total curvature of a curve equals the integral (over all line directions) of the number of critical points of projections of the curve on lines.NEWLINENEWLINEThe third chapter is devoted to planar convex sets, their support functions, and Minkowski sums. The concept of a so-called hérisson is applied, which is an envelope of a family of lines parametrized by the angle of their normal with respect to a fixed axis.NEWLINENEWLINEIn the fourth chapter, the author introduces an invariant measure on the Lie group of affine isometries of the plane, in order to derive a kinematic formula of Poincaré relating the lengths of two compact arcs and one of Blaschke for averages of Euler characteristics of intersections of a compact domain with the isometric image of another one.NEWLINENEWLINEThe fifth chapter introduces Grassmann manifolds and flag spaces. The former are used in the following chapter to establish a ``theorema egregium'' and a Gauss-Bonnet type theorem for polyhedral surfaces in \({\mathbb R}^3\), but also in the seventh chapter, where Cauchy-Crofton type formulas and exchange theorems are given for curves and surfaces in \({\mathbb R}^3\).NEWLINENEWLINEChapters 8 and 9 present results involving topological notions such as the genus of surfaces and the knottedness of curves and surfaces. Here, we mention the Chern-Lashof inequality (and the study of its equality case) and Fenchel's theorem that the total curvature of an immersed, closed curve in \({\mathbb R}^3\) is at least \(2\pi\).NEWLINENEWLINEThe tenth chapter is entitled ``3-dimensional convex bodies and related matters''. In 1901, Minkowski proved that the measure of the set of affine planes intersecting a compact convex body is proportional to the integral of the mean curvature of its boundary. Steiner termed this integral \textit{Quermassintegral}. It has the dimension of length and is generalized to non-convex surfaces. The author uses d'Ocagne's theorem to modify this notion and then to generalize Steiner's formula for the volume of thickenings of immersed compact oriented surfaces in \({\mathbb R}^3\).NEWLINENEWLINEIn Chapter 11, the author presents results from \textit{R. Langevin} and \textit{H. Rosenberg} [Comment. Math. Helv. 71, 594--616 (1996; Zbl 0880.53047)] and studies integral geometric questions on the sphere \({\mathbb S}^3\). Here, the role of lines and planes is played by geodesic circles and geodesic 2-spheres, respectively.NEWLINENEWLINEChapter 12 is devoted to the integral geometry of foliations. Here, the foliated space is one of the prototypes of constant curvature spaces, namely a Euclidean space, a sphere, or a hyperbolic space, and the dimension is two or three. As before (Chern-Lashof inequality), the particular case of so-called tight foliations is studied, when the total curvature achieves its lower bound.NEWLINENEWLINEThe main result of Chapter 13 is a Cauchy-type formula for space-like and time-like curves in the Lorentz plane.NEWLINENEWLINEThe main goal of Chapter 14 is the study of the space of two-dimensional spheres in \({\mathbb S}^3\). The corresponding group is the Möbius group, which sends spheres of \({\mathbb S}^3\) to spheres.NEWLINENEWLINEChapter 15 is entitled ``Integral geometry in hyperbolic spaces''. The Cauchy-Crofton theorems for curves and surfaces hold here mutatis mutandis, but the Chern-Lashof's inequality fails. A further Cauchy-Crofton type formula holds in the case where the lines are replaced by horocycles.NEWLINENEWLINEIn Chapter 16, entitled ``The spherical analogue of tightness'', so-called taut surfaces are studied. The Dupin cyclides are an example of such surfaces.NEWLINENEWLINEChapters 17 and 18 are devoted to conformal integral geometry. Here, conformal invariants of knots and links are addressed, and the case of foliations is studied.NEWLINENEWLINEThe author concludes ``with a chapter about integral geometry in \({\mathbb C}^2\), and an application to the study of isolated algebraic singularities.'' What follows (Chapter 20), is an ``appendix'', where basic notions for the understanding of this book are briefly explained, and also Archimedes' computation of the area between an arc of a parabola and its chord is recalled.NEWLINENEWLINEThe book ends with indices of notations, names, terms, and with an extensive bibliography.NEWLINENEWLINEAs the title suggests, this book tries to capture the major results from the very beginning of integral geometry until its very recent developments. The understanding of these results requires at least basic knowledge of the theory of topological groups, Lie groups, algebraic topology, complex analysis, and, of course, differential geometry. The 30-pages ``appendix'' (Chapter 20) is too short to provide the necessary background. There remain fundamental notions and concepts as, for instance, an immersion, an embedding, the genus, the Euler characteristic, which are neither explained or clarified, nor is any reference indicated for them. Moreover, notations are sometimes used without definition (what is \(\Phi_\epsilon\) on page 64?).NEWLINENEWLINENevertheless, by emphasizing on intuition rather than on formalism, this book gives a good impression of what integral geometry is, and it can serve both as a starting point for a gifted student and as a source of information and a guide for a researcher.
scientific article; zbMATH DE number 6567429