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Integral Geometry from Buffon to Geometers of Today

Rémi Langevin
Société Mathématique de France
Publication Date: 
Number of Pages: 
Cours Spécialisés 23
[Reviewed by
Scott Taylor
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Having just finished reading Integral geometry from Buffon to geometers of today by Rémi Langevin, I look up and see a pen lying crosswise on the hardwood floor of my living room. The image is appropriate, for the subject of integral geometry originated in Buffon’s Needle problem. In the late 18th century, Buffon proved that the probability that a “random” line segment of length \(1\) in the plane has probability \(2/\pi\) of intersecting a horizontal line of integral height.

The idea of “random” geometric objects, though appealing, has its challenges. In the 19th century, Joseph Bertrand devised three answers to the question, “What is the probability that a random chord of a circle is longer than the side of an equilateral triangle inscribed in the circle?” (See p. 13 of this book.) Depending on how the answer is calculated, Bertrand arrives at answers of \(1/3\), \(1/2\), and \(1/4\). Responding to this ambiguity, the viewpoint of geometric probability and integral geometry is that the most useful probability measure on the set of lines in \(\mathbb{R}^2\) is the one which is invariant under the Euclidean isometries of the plane. With that belief, Betrand’s question has the correct answer of \(1/2\).

With Buffon’s needle and Bertrand’s question as inspiration, we can now consider the relationship between the length of a curve in \(\mathbb{R}^2\) to the number of intersections between the curve and families of straight lines. The Cauchy-Crofton formula (page 17) states that “The length of a curve in the plane is half the average number of intersection points with lines in the plane.” With the mathematician’s penchant for generalization, it is natural to ask questions such as “What happens if we are interested in intersections with objects other than lines?”; “What if we have a curve in 3-dimensions?”; “What if we are interested in the area of a surface in 3-dimensions?”

The main purpose of Integral geometry from Buffon to geometers of today is give an exposition of how the Cauchy-Crofton theorem can be extended to many other settings. The first part of the book focuses on curves and surfaces in \(\mathbb{R}^2\) and \(\mathbb{R}^3\). The book does a good job explaining many important classical results concerning the relationship between geometric quantities such as length, area, and curvature and topological properties such as intersection numbers, Euler characteristic, and knottedness. This rather varied collection of theorems does cohere under the unifying perspective of integral geometry. Mathematicians familiar with some aspects (such as isoperimetric inequalities or the Gauss-Bonnet theorem) will enjoy seeing these results paired with lesser known results. The second half of the book is concerned with applications of integral geometry in more modern settings. Foliations, spherical geometry, hyperbolic geometry, Lorentzian geometry, and conformal geometry all merit serious attention. Many of the results discussed are due to the author and his collaborators. Researchers in topology and geometery will enjoy seeing the intellectual threads connecting classical and modern research.

The book shows its origins as a set of notes for a class. Many places in the text would benefit from more detail. Re-organizing the book slightly by moving parts of the appendix to the main body of the text would improve the exposition. Occasionally, terms and notation are introduced before they are defined. The historical comments are brief, but useful.

The background for the different sections varies but most anyone with a foundation in differential topology or geometry will find the book interesting and enlightening. Mostly only undergraduate-level differential geometry of curves and surfaces in \(\mathbb{R}^3\) is used, but sometimes Riemannian metrics and differential forms make an appearance. Without a graduate-level background in topology or geometry, the reader may find much of the material in the second half of the book unmotivated. On the other hand, with such a background the reader will immediately discern the importance of the results discussed.

As an introduction to the subjects of integral geometry and geometric probability, Langevin’s book should be read alongside the classic text by Santaló. Santaló’s text is more concerned with probabilistic questions about geometric figures in the plane, though the latter chapters deal somewhat cursorily with topics expounded in more detail and brought up-to-date in Langevin’s text.

Scott Taylor is an associate professor at Colby College. Occasionally, he feels needled by foliations.