The first thing that gets the attention of the reader is the expressive work of M. C. Esher, titled "Hand with Reflecting Sphere," that appears on the cover of the book. This is a very well written and intriguing book in which the author discusses and characterizes conformal maps and "circle/line" (sphere/hyper-plane) preserving maps on the subsets of the plane, the space, and the Euclidean spaces of higher dimension. The author also investigates the question of when a conformal map preserves convexity. Although conformal maps and circle/line preserving maps on the plane are discussed and classified in books such as Complex Analysis by Ahlfors, the book under review discusses and characterizes these maps on subsets of the Euclidean spaces of higher dimension.
A knowledge of basic concepts in topology, analysis, and differential geometry is necessary to understand most of the topics in this book. However, the author helps his readers by providing adequate review of these pre-requisites when necessary.
The author motivates his readers by proving some classical theorems, such as Miquel's Theorem and Feuerbach's Theorem, using inversion theory on the plane. In addition, the author shows how one can use inversion theory on the plane to verify the axioms of hyperbolic geometry on the Poincare model.
The author includes an original paper by C. Carathodory in which it is proved that every circle/line preserving map on the plane is a Möbius transformation. In the last chapter of the book the author discusses the question of whether inversion preserves convexity. Here he presents and proves a theorem that characterizes the points interior to a smooth closed convex surface in space that are centers of inversion preserving convexity. This theorem was first proved by the author and J.B. Wilker in a paper published in Kodai Mathematical Journal in 1982.
The reviewer believes that some aspects of inversion and conformal theory discussed in this book for the Euclidean spaces can be generalized to infinite dimensional real Hilbert spaces. For instance, a unitary linear operator on an infinite dimensional real Hilbert space is trivially a conformal map. One can also generalize these concepts to finite or infinite dimensional complex Hilbert spaces. The only difficulty is that in the case of a complex Hilbert space, the ratio of the inner product of two vectors to the product of their norms is a non-real number. However, this complexity can be resolved if one defines the (absolute) cosine of the angles between two vectors to be the ratio of absolute value of their inner product to the product of their norms. Another approach is to define the (real) cosine of the angle between two vectors to be the ratio of the real part of their inner product to the product of their norms. The reviewer and Karl Gustafson have used the concepts of absolute cosine and real cosine in two published papers on antieigenvalues of operators.
Anyone who is interested in inversion theory and conformal mapping should have this book in his personal library. This book can be used as an excellent reference book for a graduate course. It can also be used as a textbook for an advanced undergraduate course, capstone course, topics course, senior seminar or independent study.
Morteza Seddighin (firstname.lastname@example.org) is associate professor of mathematics at Indiana University East. His research interests are functional analysis and operator theory.