Preface to the English Edition

Abstract

This is the English translation of a book published in Russia in 2000. The book is a realization of our old plan to write a small book explaining the main ideas of integral geometry in the context of several simple examples, and it follows our point of view that even now certain specific examples play a bigger role in integral geometry than general results. For these simple examples we selected the classical Radon transform, its generalization suggested by F. John, hyperbolic versions of the Radon transform, and the horospherical transform for the group SL(2,C). In discussing the Radon transform, which, of course, is treated in other books, we emphasize several circumstances, which are usually not considered. One example is the projective invariance of the Radon transform. This allowed us to regard the affine Radon transform, the Minkowski–Funk transform, and the geodesic hyperbolic Radon transform as different realizations of the projective Radon transform. We also considered it important to illustrate, by simple examples, the central role played in integral geometry by the operator κ, which is responsible for the universality of explicit inversion formulas. The English edition of the book contains some modifications and corrections. In particular, we added Chapter 5 devoted to integral geometry on quadrics, or, in other words, to a conformally invariant version of the Radon transform. This approach allows us to combine the previously mentioned version of the Radon transform with the hyperbolic horospherical transform. We are grateful to Alik Shtern for his excellent translation and to Sergei Gelfand for a number of useful comments.